Method of estimating electrical parameters of an earth formation with a simplified measurement device model

ABSTRACT

Briefly, a method of estimating electrical parameters of an earth formation employs a simplified model of a measurement tool or device in transforming or normalizing data measured by the measurement tool. Electrical parameters of the earth formation, such as conductivity or dielectric constant, for example, may be estimated based on the normalized data.

RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.10/086,043 filed Feb. 28, 2002 and a continuation-in-part of U.S. patentapplication Ser. No. 09/877,383 filed on Jun. 8, 2001, now U.S. Pat. No.6,631,328, which itself is a continuation-in-part of U.S. patentapplication Ser. No. 09/608,205, filed Jun. 30, 2000, now U.S. Pat. No.6,366,858.

FIELD OF THE INVENTION

The present invention generally relates to a method of estimatingelectrical parameters of an earth formation with a simplifiedmeasurement device model.

BACKGROUND OF THE INVENTION

Typical petroleum drilling operations employ a number of techniques togather information about earth formations during and in conjunction withdrilling operations such as Wireline Logging, Measurement-While-Drilling(MWD) and Logging-While-Drilling (LWD) operations. Physical values suchas the electrical conductivity and the dielectric constant of an earthformation can indicate either the presence or absence of oil-bearingstructures near a drill hole, or “borehole.” A wealth of otherinformation that is useful for oil well drilling and production isfrequently derived from such measurements. Originally, a drill pipe anda drill bit were pulled from the borehole and then instruments wereinserted into the hole in order to collect information about down holeconditions. This technique, or “wireline logging,” can be expensive interms of both money and time. In addition, wireline data may be of poorquality and difficult to interpret due to deterioration of the regionnear the borehole after drilling. These factors lead to the developmentof Logging-While-Drilling (LWD). LWD operations involve collecting thesame type of information as wireline logging without the need to pullthe drilling apparatus from the borehole. Since the data are taken whiledrilling, the measurements are often more representative of virginformation conditions because the near-borehole region often deterioratesover time after the well is drilled. For example, the drilling fluidoften penetrates or invades the rock over time, making it more difficultto determine whether the fluids observed within the rock are naturallyoccurring or drilling induced. Data acquired while drilling are oftenused to aid the drilling process. For example, MWD/LWD data can help adriller navigate the well so that the borehole is ideally positionedwithin an oil bearing structure. The distinction between LWD and MWD isnot always obvious, but MWD usually refers to measurements taken for thepurpose of drilling the well (such as navigation) whereas LWD isprincipally for the purpose of estimating the fluid production from theearth formation. These terms will hereafter be used synonymously andreferred to collectively as “MWD/LWD.”

In wireline logging, wireline induction measurements are commonly usedto gather information used to calculate the electrical conductivity, orits inverse resistivity. See for example U.S. Pat. No. 5,157,605. Adielectric wireline tool is used to determine the dielectric constantand/or resistivity of an earth formation. This is typically done usingmeasurements which are sensitive to the volume near the borehole wall.See for example U.S. Pat. No. 3,944,910. In MWD/LWD, a MWD/LWDresistivity tool is typically employed. Such devices are often called“propagation resistivity” or “wave resistivity” tools, and they operateat frequencies high enough that the measurement is sensitive to thedielectric constant under conditions of either high resistivity or alarge dielectric constant. See for example U.S. Pat. Nos. 4,899,112 and4,968,940. In MWD applications, resistivity measurements may be used forthe purpose of evaluating the position of the borehole with respect toboundaries of the reservoir such as with respect to a nearby shale bed.The same resistivity tools used for LWD may also used for MWD; but, inLWD, other formation evaluation measurements including density andporosity are typically employed.

For purposes of this disclosure, the terms “resistivity” and“conductivity” will be used interchangeably with the understanding thatthey are inverses of each other and the measurement of either can beconverted into the other by means of simple mathematical calculations.The terms “depth,” “point(s) along the borehole,” and “distance alongthe borehole axis” will also be used interchangeably. Since the boreholeaxis may be tilted with respect to the vertical, it is sometimesnecessary to distinguish between the vertical depth and distance alongthe borehole axis. Should the vertical depth be referred to, it will beexplicitly referred to as the “vertical depth.”

Typically, the electrical conductivity of an earth formation is notmeasured directly. It is instead inferred from other measurements eithertaken during (MWD/LWD) or after (Wireline Logging) the drillingoperation. In typical embodiments of MWD/LWD resistivity devices, thedirect measurements are the magnitude and the phase shift of atransmitted electrical signal traveling past a receiver array. See forexample U.S. Pat. Nos. 4,899,112, 4,968,940, or 5,811,973. In commonlypracticed embodiments, the transmitter emits electrical signals offrequencies typically between four hundred thousand and two millioncycles per second (0.4–2.0 MHz). Two induction coils spaced along theaxis of the drill collar having magnetic moments substantially parallelto the axis of the drill collar typically comprise the receiver array.The transmitter is typically an induction coil spaced along the axis ofa drill collar from the receiver with its magnetic moment substantiallyparallel to the axis of the drill collar. A frequently used mode ofoperation is to energize the transmitter for a long enough time toresult in the signal being essentially a continuous wave (only afraction of a second is needed at typical frequencies of operation). Themagnitude and phase of the signal at one receiving coil is recordedrelative to its value at the other receiving coil. The magnitude isoften referred to as the attenuation, and the phase is often called thephase shift. Thus, the magnitude, or attenuation, and the phase shift,or phase, are typically derived from the ratio of the voltage at onereceiver antenna relative to the voltage at another receiver antenna.

Commercially deployed MWD/LWD resistivity measurement systems usemultiple transmitters; consequently, attenuation and phase-basedresistivity values can be derived independently using each transmitteror from averages of signals from two or more transmitters. See forexample U.S. Pat. No. 5,594,343.

As demonstrated in U.S. Pat. Nos. 4,968,940 and 4,899,112, a very commonmethod practiced by those skilled in the art of MWD/LWD for determiningthe resistivity from the measured data is to transform the dielectricconstant into a variable that depends on the resistivity and then toindependently convert the phase shift and attenuation measurements totwo separate resistivity values. A key assumption implicitly used inthis practice is that each measurement senses the resistivity within thesame volume that it senses the dielectric constant. This currentlypracticed method may provide significantly incorrect resistivity values,even in virtually homogeneous earth formations; and the errors may beeven more severe in inhomogeneous formations. A MWD/LWD tool typicallytransmits a 2 MHz signal (although frequencies as low as 0.4 MHz aresometimes used). This frequency range is high enough to createdifficulties in transforming the raw attenuation and phase measurementsinto accurate estimates of the resistivity and/or the dielectricconstant. For example, the directly measured values are not linearlydependent on either the resistivity or the dielectric constant (thisnonlinearity, known to those skilled in the art as “skin-effect,” alsolimits the penetration of the fields into the earth formation). Inaddition, it is useful to separate the effects of the dielectricconstant and the resistivity on the attenuation and phase measurementsgiven that both the resistivity and the dielectric constant typicallyvary spatially within the earth formation. If the effects of both ofthese variables on the measurements are not separated, the estimate ofthe resistivity can be corrupted by the dielectric constant, and theestimate of the dielectric constant can be corrupted by the resistivity.Essentially, the utility of separating the effects is to obtainestimates of one parameter that do not depend on (are independent of)the other parameter. A commonly used current practice relies on assuminga correlative relationship between the resistivity and dielectricconstant (i.e., to transform the dielectric constant into a variablethat depends on the resistivity) and then calculating resistivity valuesindependently from the attenuation and phase shift measurements that areconsistent the correlative relationship. Differences between theresistivity values derived from corresponding phase and attenuationmeasurements are then ascribed to spatial variations (inhomogeneities)in the resistivity over the sensitive volume of the phase shift andattenuation measurements. See for example U.S. Pat. Nos. 4,899,112 and4,968,940. An implicit and instrumental assumption in this method isthat the attenuation measurement senses both the resistivity anddielectric constant within the same volume, and that the phase shiftmeasurement senses both variables within the same volume (but not thesame volume as the attenuation measurement). See for example U.S. Pat.Nos. 4,899,112 and 4,968,940. These assumptions facilitate theindependent determination of a resistivity value from a phasemeasurement and another resistivity value from an attenuationmeasurement. However, the implicit assumption mentioned above is nottrue; so, the results determined using such algorithms are questionable.

Another technique for determining the resistivity and/or dielectricconstant is to assume a model for the measurement apparatus in, forexample, a homogeneous medium (no spatial variation in either theresistivity or dielectric constant) and then to determine values for theresistivity and dielectric constant that cause the model to agree withthe measured phase shift and attenuation data. The resistivity anddielectric constant determined by the model are then correlated to theactual parameters of the earth formation. This method is thought to bevalid only in a homogeneous medium because of the implicit assumptionmentioned in the above paragraph. A recent publication by P. T. Wu, J.R. Lovell, B. Clark, S. D. Bonner, and J. R. Tabanou entitled“Dielectric-Independent 2-MHz Propagation Resistivities” (SPE 56448,1999) (hereafter referred to as “Wu”) demonstrates that such assumptionsare used by those skilled in the art. For example, Wu states: “Onefundamental assumption in the computation of Rex is an uninvadedhomogeneous formation. This is because the phase shift and attenuationinvestigate slightly different volumes.” Abandoning the falseassumptions applied in this practice results in estimates of oneparameter (i.e., the resistivity or dielectric constant) that have nonet sensitivity to the other parameter. This desirable and previouslyunknown property of the results is very useful because earth formationsare commonly inhomogeneous.

Wireline dielectric measurement tools commonly use electrical signalshaving frequencies in the range 20 MHz–1.1 GHz. In this range, theskin-effect is even more severe, and it is even more useful to separatethe effects of the dielectric constant and resistivity. Those skilled inthe art of dielectric measurements have also falsely assumed that ameasurement (either attenuation or phase) senses both the resistivityand dielectric constant within the same volume. The design of themeasurement equipment and interpretation of the data both reflect this.See for example U.S. Pat. Nos. 4,185,238 and 4,209,747.

Wireline induction measurements are typically not attenuation and phase,but instead the real (R) and imaginary (X) parts of the voltage across areceiver antenna which consists of several induction coils in electricalseries. For the purpose of this disclosure, the R-signal for a wirelineinduction measurement corresponds to the phase measurement of a MWD/LWDresistivity or wireline dielectric tool, and the X-signal for a wirelineinduction measurement corresponds to the attenuation measurement of aMWD/LWD resistivity or wireline dielectric device. Wireline inductiontools typically operate using electrical signals at frequencies from8–200 kHz (most commonly at approximately 20 kHz). This frequency rangeis too low for significant dielectric sensitivity in normallyencountered cases; however, the skin-effect can corrupt the wirelineinduction measurements. As mentioned above, the skin-effect shows up asa non-linearity in the measurement as a function of the formationconductivity, and also as a dependence of the measurement sensitivityvalues on the formation conductivity. Estimates of the formationconductivity from wireline induction devices are often derived from dataprocessing algorithms which assume the tool response function is thesame at all depths within the processing window. The techniquesdisclosed in the commonly-assigned U.S. patent application, Ser. No.09/877,383, entitled “Method of Determining Resistivity of an EarthFormation with Phase Resistivity Evaluation Based on a Phase ShiftMeasurement and Attenuation Resistivity Evaluation Based on anAttenuation Measurement and the Phase Shift Measurement,” can be appliedto wireline induction measurements for the purpose of derivingresistivity values without assuming the tool response function is thesame at all depths within the processing window as is done in U.S. Pat.No. 5,157,605. In order to make such an assumption, a backgroundconductivity, σ, that applies for the data within the processing windowis commonly used. Practicing an embodiment as disclosed in the abovereferenced, commonly assigned U.S. Patent Application reduces thedependence of the results on the accuracy of the estimates for thebackground parameters because the background parameters are not requiredto be the same at all depths within the processing window. Anotheradvantage is reduction of the need to perform steps to correct wirelineinduction data for the skin effect. Such parameterization as disclosedin the above referenced, commonly assigned U.S. Patent Application,however, has not fully taken into account complications such as invadedzones and frequency dispersion.

SUMMARY OF THE INVENTION

Briefly, a method of estimating electrical parameters of an earthformation employs a simplified model of a measurement tool or device intransforming or normalizing data measured by the measurement tool.Electrical parameters of the earth formation, such as conductivity ordielectric constant, for example, may be estimated based on thenormalized data.

The foregoing has outlined rather broadly the features and technicaladvantages of the present invention in order that the detaileddescription of the invention that follows may be better understood.Additional features and advantages of the invention will be describedhereinafter which form the subject of the claims of the invention. Itshould be appreciated by those skilled in the art that the conceptionand the specific embodiment disclosed may be readily utilized as a basisfor modifying or designing other structures for carrying out the samepurposes of the present invention. It should be also be realized bythose skilled in the art that such equivalent constructions do notdepart from the spirit and scope of the invention as set forth in theappended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention, and theadvantages thereof, reference is now made to the following descriptionstaken in conjunction with the accompanying drawings, in which:

FIG. 1 is a plot of multiple laboratory measurements on rock samplesrepresenting the relationship between the conductivity and thedielectric constant in a variety of geological media;

FIG. 2 illustrates the derivation of a sensitivity function in relationto an exemplary one-transmitter, one-receiver MWD/LWD resistivity tool;

FIG. 3 illustrates an exemplary one-transmitter, two-receiver MWD/LWDtool commonly referred to as an uncompensated measurement device;

FIG. 4 illustrates an exemplary two-transmitter, two-receiver MWD/LWDtool, commonly referred to as a compensated measurement device;

FIGS. 5 a, 5 b, 5 c and 5 d are exemplary sensitivity function plots forDeep and Medium attenuation and phase shift measurements;

FIGS. 6 a, 6 b, 6 c and 6 d are plots of the sensitivity functions forthe Deep and Medium measurements of FIGS. 5 a, 5 b, 5 c and 5 drespectively transformed according to the techniques of a disclosedembodiment;

FIG. 7 is a portion of a table of background medium values and integralvalues employed in a disclosed embodiment;

FIG. 8 is a plot of attenuation and phase as a function of resistivityand dielectric constant;

FIG. 9 is a flowchart of a process that implements the techniques of adisclosed embodiment;

FIG. 10 is a table of exemplary data values for a medium spaced 2 MHzmeasurement described in conjunction with FIG. 7;

FIG. 11 is a table of exemplary data values for a medium spaced 2 MHzmeasurement illustrating the difference between measured attenuation andphase-shift values and corresponding point-dipole attenuation andphase-shift values for a fixed resistivity as a function of thedielectric constant;

FIG. 12 is a plot showing the difference between measured (mandrel) andpoint-dipole values as a function of both the resistivity and dielectricconstant for the medium spaced 2 MHz measurement; and

FIG. 13 is a plot showing the performance of the extended approximationsthat facilitate use of rapidly-evaluated models.

DETAILED DESCRIPTION

Some of the disclosed embodiments are relevant to both wirelineinduction and dielectric applications, as well asMeasurement-While-Drilling and Logging-While-Drilling (MWD/LWD)applications. Turning now to the figures, FIG. 1 is a plot ofmeasurements of the conductivity and dielectric constant determined bylaboratory measurements on a variety of rock samples from differentgeological environments. The points 121 through 129 represent measuredvalues of conductivity and dielectric constant (electrical parameters)for carbonate and sandstone earth formations. For instance, the point126 represents a sample with a conductivity value of 0.01 (10⁻²) siemensper meter (S/m) and a relative dielectric constant of approximately 22.It should be noted that both the conductivity scale and the dielectricscale are logarithmic scales; so, the data would appear to be much morescattered if they were plotted on linear scales.

The maximum boundary 111 indicates the maximum dielectric constantexpected to be observed at each corresponding conductivity. In a similarfashion, the minimum boundary 115 represents the minimum dielectricconstant expected to be observed at each corresponding conductivity. Thepoints 122 through 128 represent measured values that fall somewhere inbetween the minimum boundary 115 and the maximum boundary 111. A medianline 113 is a line drawn so that half the points, or points 121 through124 are below the median line 113 and half the points, or points 126through 129 are above the median line 113. The point 125 falls right ontop of the median line 113.

An elemental measurement between a single transmitting coil 205 and asingle receiving coil 207 is difficult to achieve in practice, but it isuseful for describing the sensitivity of the measurement to variationsof the conductivity and dielectric constant within a localized volume225 of an earth formation 215. FIG. 2 illustrates in more detailspecifically what is meant by the term “sensitivity function,” alsoreferred to as a “response function” or “geometrical factors.”Practitioners skilled in the art of wireline logging,Measurement-While-Drilling (MWD) and Logging-While-Drilling (LWD) arefamiliar with how to generalize the concept of a sensitivity function toapply to realistic measurements from devices using multiple transmittingand receiving antennas. Typically a MWD/LWD resistivity measurementdevice transmits a signal using a transmitter coil and measures thephase and magnitude of the signal at one receiver antenna 307 relativeto the values of the phase and the magnitude at another receiver antenna309 within a borehole 301 (FIG. 3). These relative values are commonlyreferred to as the phase shift and attenuation. It should be understoodthat one way to represent a complex signal with multiple components isas a phasor signal.

Sensitivity Functions

FIG. 2 illustrates an exemplary single transmitter, single receiverMWD/LWD resistivity tool 220 for investigating an earth formation 215. Ametal shaft, or “mandrel,” 203 is incorporated within the drill string(the drill string is not shown, but it is a series of pipes screwedtogether with a drill bit on the end), inserted into the borehole 201,and employed to take measurements of an electrical signal thatoriginates at a transmitter 205 and is sensed at a receiver 207. Themeasurement tool is usually not removed from the well until the drillstring is removed for the purpose of changing drill bits or becausedrilling is completed. Selected data from the tool are telemetered tothe surface while drilling. All data are typically recorded in memorybanks for retrieval after the tool is removed from the borehole 201.Devices with a single transmitter and a single receiver are usually notused in practice, but they are helpful for developing concepts such asthat of the sensitivity function. Schematic drawings of simple,practical apparatuses are shown in FIGS. 3 and 4.

In a wireline operation, the measurement apparatus is connected to acable (known as a wireline), lowered into the borehole 201, and data areacquired. This is done typically after the drilling operation isfinished. Wireline induction tools measure the real (R) and imaginary(X) components of the receiver 207 signal. The R and X-signalscorrespond to the phase shift and attenuation measurements respectively.In order to correlate the sensitivity of the phase shift and attenuationmeasurements to variations in the conductivity and dielectric constantof the earth formation 215 at different positions within the earthformation, the conductivity and dielectric constant within a smallvolume P 225 are varied. For simplicity, the volume P 225 is a solid ofrevolution about the tool axis (such a volume is called atwo-dimensional volume). The amount the phase and attenuationmeasurements change relative to the amount the conductivity anddielectric constant changed within P 225 is essentially the sensitivity.The sensitivity function primarily depends on the location of the pointP 225 relative to the locations of the transmitter 205 and receiver 207,on the properties of the earth formation 215, and on the excitationfrequency. It also depends on other variables such as the diameter andcomposition of the mandrel 203, especially when P is near the surface ofthe mandrel 203.

Although the analysis is carried out in two-dimensions, the importantconclusions regarding the sensitive volumes of phase shift andattenuation measurements with respect to the conductivity and dielectricconstant hold in three-dimensions. Consequently, the scope of thisapplication is not limited to two-dimensional cases. This is discussedmore in a subsequent section entitled, “ITERATIVE FORWARD MODELING ANDDIPPING BEDS.”

The sensitivity function can be represented as a complex number having areal and an imaginary part. In the notation used below, S, denotes acomplex sensitivity function, and its real part is S′, and its imaginarypart is S″. Thus, S=S′+iS″, in which the imaginary number i=√{squareroot over (−1)}. The quantities S′ and S″ are commonly referred to asgeometrical factors or response functions. The volume P 225 is located adistance ρ in the radial direction from the tool's axis and a distance zin the axial direction from the receiver 206. S′ represents thesensitivity of attenuation to resistivity and the sensitivity of phaseshift to dielectric constant. Likewise, S″ represents the sensitivity ofattenuation to dielectric constant and the sensitivity of phase shift toresistivity. The width of the volume P 225 is Δρ 211 and the height ofthe volume P 225 is Δz 213. The quantity S′, or the sensitivity ofattenuation to resistivity, is calculated by determining the effect achange in the conductivity (reciprocal of resistivity) in volume P 225from a prescribed background value has on the attenuation of a signalbetween the transmitter 205 and the receiver 207, assuming thebackground conductivity value is otherwise unperturbed within the entireearth formation 215. In a similar fashion, S″, or the sensitivity of thephase to the resistivity, is calculated by determining the effect achange in the conductivity value in the volume P 225 from an assumedbackground conductivity value has on the phase of the signal between thetransmitter 205 and the receiver 207, assuming the background parametersare otherwise unperturbed within the earth formation 215. Alternatively,one could determine S′ and S″ by determining the effect a change in thedielectric constant within the volume P 225 has on the phase andattenuation, respectively. When the sensitivities are determined byconsidering a perturbation to the dielectric constant value within thevolume P 225, it is apparent that the sensitivity of the attenuation tochanges in the dielectric constant is the same as the sensitivity of thephase to the conductivity. It is also apparent that the sensitivity ofthe phase to the dielectric constant is the same as the sensitivity ofthe attenuation to the conductivity. By simultaneously considering thesensitivities of both the phase and attenuation measurement to thedielectric constant and to the conductivity, the Applicant shows apreviously unknown relationship between the attenuation and phase shiftmeasurements and the conductivity and dielectric constant values. Byemploying this previously unknown relationship, the Applicant providestechniques that produce better estimates of both the conductivity andthe dielectric constant values than was previously available from thosewith skill in the art. The sensitivity functions S′ and S″ and theirrelation to the subject matter of the Applicant's disclosure isexplained in more detail below in conjunction with FIGS. 5 a–d and FIGS.6 a–d.

In the above, sensitivities to the dielectric constant were referred to.Strictly speaking, the sensitivity to the radian frequency ω times thedielectric constant should have been referred to. This distinction istrivial to those skilled in the art.

In FIG. 2, if the background conductivity (reciprocal of resistivity) ofthe earth formation 215 is σ₀ and the background dielectric constant ofthe earth formation 215 is ∈₀, then the ratio of the receiver 207voltage to the transmitter 205 current in the background medium can beexpressed as Z_(RT) ⁰, where R stands for the receiver 207 and T standsfor the transmitter 205. Hereafter, a numbered subscript or superscriptsuch as the ‘0’ is merely used to identify a specific incidence of thecorresponding variable or function. If an exponent is used, the variableor function being raised to the power indicated by the exponent will besurrounded by parentheses and the exponent will be placed outside theparentheses. For example (L₁)³ would represent the variable L₁ raised tothe third power.

When the background conductivity σ₀ and/or dielectric constant ∈₀ arereplaced new values σ₁, and/or ∈₁ in the volume P 225, the ratio betweenthe receiver 207 voltage to the transmitter 205 current is representedby Z_(RT) ¹. Using the same nomenclature, a ratio between a voltage at ahypothetical receiver placed in the volume P 225 and the current at thetransmitter 205 can be expressed as Z_(PT) ⁰. In addition, a ratiobetween the voltage at the receiver 207 and a current at a hypotheticaltransmitter in the volume P 225 can be expressed as Z_(RP) ⁰. Using theBorn approximation, it can be shown that,

$\frac{Z_{RT}^{1}}{Z_{RT}^{0}} = {1 + {{S\left( {T,R,P} \right)}\Delta\;\overset{\sim}{\sigma}\;\Delta\;\rho\;\Delta\; z}}$

where the sensitivity function, defined as S(T,R,P), is

${S\left( {T,R,P} \right)} = {- \frac{Z_{RP}^{0}Z_{PT}^{0}}{2{\pi\rho}\; Z_{RT}^{0}}}$

in which Δ{tilde over (σ)}={tilde over (σ)}₁−{tilde over(σ)}₀=(σ₁−σ₀)+iω(∈₁−∈₀), and the radian frequency of the transmittercurrent is ω=2πƒ. A measurement of this type, in which there is just onetransmitter 205 and one receiver 207, is defined as an “elemental”measurement. It should be noted that the above result is also valid ifthe background medium parameters vary spatially within the earthformation 215. In the above equations, both the sensitivity functionS(T, R, P) and the perturbation Δ{tilde over (σ)} are complex-valued.Some disclosed embodiments consistently treat the measurements, theirsensitivities, and the parameters to be estimated as complex-valuedfunctions. This is not done in the prior art.

The above sensitivity function of the form S(T, R, P) is referred to asa 2-D (or two-dimensional) sensitivity function because the volume ΔρΔzsurrounding the point P 225, is a solid of revolution about the axis ofthe tool 201. Because the Born approximation was used, the sensitivityfunction S depends only on the properties of the background mediumbecause it is assumed that the same field is incident on the point P(ρ,z) even though the background parameters have been replaced by {tildeover (σ)}₁.

FIG. 3 illustrates an exemplary one-transmitter, two-receiver MWD/LWDresistivity measurement apparatus 320 for investigating an earthformation 315. Due to its configuration, the tool 320 is defined as an“uncompensated” device and collects uncompensated measurements from theearth formation 315. For the sake of simplicity, a borehole is notshown. This measurement tool 320 includes a transmitter 305 and tworeceivers 307 and 309, each of which is incorporated into a metalmandrel 303. Typically, the measurement made by such a device is theratio of the voltages at receivers 307 and 309. In this example, usingthe notation described above in conjunction with FIG. 2, the sensitivityfunction S(T, R, R′, P) for the uncompensated device can be shown to bethe difference between the elemental sensitivity functions S(T,R,P) andS(T, R′, P), where T represents the transmitter 305, R represents thereceiver 307, R′ represents the receiver 309, and P represents a volume(not shown) similar to the volume P 225 of FIG. 2.

For wireline induction measurements, the voltage at the receiver R issubtracted from the voltage at the receiver R′, and the position andnumber of turns of wire for R are commonly chosen so that the differencein the voltages at the two receiver antennas is zero when the tool is ina nonconductive medium. For MWD/LWD resistivity and wireline dielectricconstant measurements, the voltage at the receiver R, or V_(R), and thevoltage at the receiver R′, or V_(R′), are examined as the ratioV_(R)/V_(R′). In either case, it can be shown thatS(T,R,R′,P)=S(T,R,P)−S(T,R′,P).

The sensitivity for an uncompensated measurement is the differencebetween the sensitivities of two elemental measurements such as S(T,R,P)and S(T, R′, P) calculated as described above in conjunction with FIG.2.

FIG. 4 illustrates an exemplary two-transmitter, two-receiver MWD/LWDresistivity tool 420. Due to its configuration (transmitters beingdisposed symmetrically), the tool 420 is defined as a “compensated” tooland collects compensated measurements from an earth formation 415. Thetool 420 includes two transmitters 405 and 411 and two receivers 407 and409, each of which is incorporated into a metal mandrel or collar 403.Each compensated measurement is the geometric mean of two correspondinguncompensated measurements. In other words, during a particulartimeframe, the tool 420 performs two uncompensated measurements, oneemploying transmitter 405 and the receivers 407 and 409 and the otheremploying the transmitter 411 and the receivers 409 and 407. These twouncompensated measurements are similar to the uncompensated measurementdescribed above in conjunction with FIG. 3. The sensitivity function Sof the tool 420 is then defined as the arithmetic average of thesensitivity functions for each of the uncompensated measurements.Another way to describe this relationship is with the following formula:

${S\left( {T,R,R^{\prime},T^{\prime},P} \right)} = {\frac{1}{2}\left\lbrack {{S\left( {T,R,R^{\prime},P} \right)} + {S\left( {T^{\prime},R^{\prime},R,P} \right)}} \right\rbrack}$

where T represents transmitter 405, T′ represents transmitter 411, Rrepresents receiver 407, R′ represents receiver 409 and P represents asmall volume of the earth formation similar to 225 (FIG. 2).

The techniques of the disclosed embodiments are explained in terms of acompensated tool such as the tool 420 and compensated measurements suchas those described in conjunction with FIG. 4. However, it should beunderstood that the techniques also apply to uncompensated tools such asthe tool 320 and uncompensated measurements described above inconjunction with FIG. 3 and elemental tools such as the tool 220 andelemental measurements such as those described above in conjunction withFIG. 2. In addition, the techniques are applicable for use in a wirelinesystem, a system that may not incorporate its transmitters and receiversinto a metal mandrel, but may rather affix a transmitter and a receiverto a tool made of a non-conducting material such as fiberglass. Thewireline induction frequency is typically too low for dielectric effectsto be significant. Also typical for wireline induction systems is toselect the position and number of turns of groups of receiver antennasso that there is a null signal in a nonconductive medium. When this isdone, Z_(RT) ⁰=0 if {tilde over (σ)}₀=0. As a result, it is necessary tomultiply the sensitivity and other quantities by Z_(RT) ⁰ to use theformulation given here in such cases.

The quantity ZR_(RT) ¹/Z_(RT) ⁰ can be expressed as a complex numberwhich has both a magnitude and a phase (or alternatively real andimaginary parts). To a good approximation, the raw attenuation value(which corresponds to the magnitude) is:

${{\frac{Z_{RT}^{1}}{Z_{RT}^{0}}} \approx {{Re}\left\lbrack \frac{Z_{RT}^{1}}{Z_{RT}^{0}} \right\rbrack}} = {{1 + {{{Re}\left\lbrack {{S\left( {T,R,P} \right)}\Delta\;\overset{\sim}{\sigma}} \right\rbrack}\Delta\;{\rho\Delta}\; z}} = {1 + {\left\lbrack {{S^{\prime}{\Delta\sigma}} - {S^{''}\omega\;\Delta\; ɛ}} \right\rbrack\Delta\;{\rho\Delta}\; z}}}$

where the function Re[·] denotes the real part of its argument. Also, toa good approximation, the raw phase shift value is:

${{{phase}\left( \frac{Z_{RT}^{1}}{Z_{RT}^{0}} \right)} \approx {{Im}\left\lbrack \frac{Z_{RT}^{1}}{Z_{RT}^{0}} \right\rbrack}} = {{{{Im}\left\lbrack {{S\left( {T,R,P} \right)}\Delta\;\overset{\sim}{\sigma}} \right\rbrack}\Delta\;{\rho\Delta}\; z} = {\left\lbrack {{S^{''}\Delta\;\sigma} + {S^{\prime}{\omega\Delta ɛ}}} \right\rbrack\Delta\;\rho\;\Delta\; z}}$

in which Im[·] denotes the imaginary part of its argument, S(T, R,P)=S′+iS″, Δσ=σ₁−σ₀, and Δ∈=∈₁−∈₀. For the attenuation measurement, S′is the sensitivity to the resistivity and S″ is the sensitivity to thedielectric constant. For the phase shift measurement, S′ is thesensitivity to the dielectric constant and S″ is the sensitivity to theresistivity. This is apparent because S′ is the coefficient of Δσ in theequation for attenuation, and it is also the coefficient of ωΔ∈ in theequation for the phase shift. Similarly, S″ is the coefficient of Δσ inthe equation for the phase shift, and it is also the coefficient for−ωΔ∈ in the equation for attenuation. This implies that the attenuationmeasurement senses the resistivity in the same volume as the phase shiftmeasurement senses the dielectric constant and that the phase shiftmeasurement senses the resistivity in the same volume as the attenuationmeasurement senses the dielectric constant. In the above, we havereferred to sensitivities to the dielectric constant. Strictly speaking,the sensitivity to the radian frequency ω times the dielectric constantΔ∈ should have been referred to. This distinction is trivial to thoseskilled in the art.

The above conclusion regarding the volumes in which phase andattenuation measurements sense the resistivity and dielectric constantfrom Applicant's derived equations also follows from a well known resultfrom complex variable theory known in that art as the Cauchy-Reimannequations. These equations provide the relationship between thederivatives of the real and imaginary parts of an analytic complexfunction with respect to the real and imaginary parts of the function'sargument.

FIGS. 5 a, 5 b, 5 c and 5 d can best be described and understoodtogether. In all cases, the mandrel diameter is 6.75 inches, thetransmitter frequency is 2 MHz, and the background medium ischaracterized by a conductivity of σ₀=0.01 S/m and a relative dielectricconstant of ∈₀=10. The data in FIGS. 5 a and 5 c labeled “MediumMeasurement” are for a compensated type of design shown in FIG. 4. Theexemplary distances between transmitter 405 and receivers 407 and 409are 20 and 30 inches, respectively. Since the tool is symmetric, thedistances between transmitter 411 and receivers 409 and 407 are 20 and30 inches, respectively. The data in FIGS. 5 b and 5 d labeled “DeepMeasurement” are also for a compensated tool as shown in FIG. 4, butwith exemplary transmitter-receiver spacings of 50 and 60 inches. Eachplot shows the sensitivity of a given measurement as a function ofposition within the formation. The term sensitive volume refers to theshape of each plot as well as its value at any point in the formation.The axes labeled “Axial Distance” refer to the coordinate along the axisof the tool with zero being the geometric mid-point of the antenna array(halfway between receivers 407 and 409) to a given point in theformation. The axes labeled “Radial Distance” refer to the radialdistance from the axis of the tool to a given point in the formation.The value on the vertical axis is actually the sensitivity value for theindicated measurement. Thus, FIG. 5 a is a plot of a sensitivityfunction that illustrates the sensitivity of the “Medium” phase shiftmeasurement in relation to changes in the resistivity as a function ofthe location of the point P 225 in the earth formation 215 (FIG. 2). Ifthe measurement of phase shift changes significantly in response tochanging the resistivity from its background value, then phase shift isconsidered relatively sensitive to the resistivity at the point P 225.If the measurement of phase shift does not change significantly inresponse to changing the resistivity, then the phase shift is consideredrelatively insensitive at the point P 225. Based upon the relationshipdisclosed herein, FIG. 5 a also illustrates the sensitivity of the“Medium” attenuation measurement in relation to changes in dielectricconstant values. Note that the dimensions of the sensitivity on thevertical axes is ohms per meter (Ω/m) and distances on the horizontalaxes are listed in inches. In a similar fashion, FIG. 5 b is a plot ofthe sensitivity of the attenuation measurement to the resistivity. Basedon the relationship disclosed herein, FIG. 5 b is also the sensitivityof a phase shift measurement to a change in the dielectric constant.FIGS. 5 b and 5 d have the same descriptions as FIGS. 5 a and 5 c,respectively, but FIGS. 5 b and 5 d are for the “Deep Measurement” withthe antenna spacings described above.

Note that the shape of FIG. 5 a is very dissimilar to the shape of FIG.5 c. This means that the underlying measurements are sensitive to thevariables in different volumes. For example, the Medium phase shiftmeasurement has a sensitive volume characterized by FIG. 5 a for theresistivity, but this measurement has the sensitive volume shown in FIG.5 c for the dielectric constant. FIGS. 5 a–5 d illustrate that for aparticular measurement the surface S″ is more localized than the surfaceS′ such that the sensitive volume associated with the surface S′substantially encloses the sensitive volume associated with S″ for thecorresponding measurement. As discussed below, it is possible totransform an attenuation and a phase shift measurement to a complexnumber which has the following desirable properties: 1) its real part issensitive to the resistivity in the same volume that the imaginary partis sensitive to the dielectric constant; 2) the real part has no netsensitivity to the dielectric constant; and, 3) the imaginary part hasno net sensitivity to the resistivity. In addition, the transformationis generalized to accommodate multiple measurements acquired at multipledepths. The generalized method can be used to produce independentestimates of the resistivity and dielectric constant within a pluralityof volumes within the earth formation.

Transformed Sensitivity Functions and Transformation of the Measurements

For simplicity, the phase shift and attenuation will not be used.Hereafter, the real and imaginary parts of measurement will be referredto instead. Thus,w=w′+iw″w′=(10)^(dB/20)×cos(θ)w″=(10)^(dB/20)×sin(θ)where w′ is the real part of w, w″ is the imaginary part of w, i is thesquare root of the integer −1, dB is the attenuation in decibels, and θis the phase shift in radians.

The equations that follow can be related to the sensitivity functionsdescribed above in conjunction with FIG. 2 by defining variablesw₁=Z_(RT) ¹ and w₀=Z_(RT) ⁰. The variable w₁ denotes an actual toolmeasurement in the earth formation 215. The variable w₀ denotes theexpected value for the tool measurement in the background earthformation 215. For realistic measurement devices such as those describedin FIGS. 3 and 4, the values for w₁ and w₀ would be the voltage ratiosdefined in the detailed description of FIGS. 3 and 4. In one embodiment,the parameters for the background medium are determined and then used tocalculate value of w₀ using a mathematical model to evaluate the toolresponse in the background medium. One of many alternative methods todetermine the background medium parameters is to estimate w₀ directlyfrom the measurements, and then to determine the background parametersby correlating w₀ to a model of the tool in the formation which has thebackground parameters as inputs.

As explained in conjunction with FIG. 2, the sensitivity function Srelates the change in the measurement to a change in the mediumparameters such as resistivity and dielectric constant within a smallvolume 225 of the earth formation 215 at a prescribed location in theearth formation 225, or background medium. A change in measurements dueto small variations in the medium parameters at a range of locations canbe calculated by integrating the responses from each such volume in theearth formation 215. Thus, if Δ{tilde over (σ)} is defined for a largenumber of points ρ, z, thenZ _(RT) ¹ =Z _(RT) ⁰(1+I[SΔ{tilde over (σ)}])

in which I is a spatial integral function further defined as

I[F] = ∫_(−∞)^(+∞)𝕕z∫₀^(+∞)𝕕ρ F(ρ, z)

where F is a complex function.

Although the perturbation from the background medium, Δ{tilde over (σ)}is a function of position, parameters of a hypothetical, equivalenthomogeneous perturbation (meaning that no spatial variations are assumedin the difference between the resistivity and dielectric constant andvalues for both of these parameters in the background medium) can bedetermined by assuming the perturbation is not a function of positionand then solving for it. Thus,Δ{circumflex over (σ)}I[S]=I[SΔ{tilde over (σ)}]

where Δ{circumflex over (σ)} represents the parameters of the equivalenthomogeneous perturbation. From the previous equations, it is clear that

${\Delta\;\hat{\sigma}} = {\frac{I\left\lbrack {S\;\Delta\;\overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\;\Delta\;\overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}\left( {\frac{w_{1}}{w_{0}} - 1} \right)}}}$and $\hat{S} = \frac{S}{I\lbrack S\rbrack}$

where Δ{circumflex over (σ)} is the transformed measurement (it isunderstood that Δ{circumflex over (σ)} is also the equivalenthomogeneous perturbation and that the terms transformed measurement andequivalent homogeneous perturbation will be used synonymously), Ŝ is thesensitivity function for the transformed measurement, and Ŝ will bereferred to as the transformed sensitivity function. In the above, w₁ isthe actual measurement, and w₀ is the value assumed by the measurementin the background medium. An analysis of the transformed sensitivityfunction Ŝ, shows that the transformed measurements have the followingproperties: 1) the real part of Δ{circumflex over (σ)} is sensitive tothe resistivity in the same volume that its imaginary part is sensitiveto the dielectric constant; 2) the real part of Δ{circumflex over (σ)}has no net sensitivity to the dielectric constant; and, 3) the imaginarypart of Δ{circumflex over (σ)} has no net sensitivity to theresistivity. Details of this analysis will be given in the next fewparagraphs.

The techniques of the disclosed embodiment can be further refined byintroducing a calibration factor c (which is generally a complex numberthat may depend on the temperature of the measurement apparatus andother environmental variables) to adjust for anomalies in the physicalmeasurement apparatus. In addition, the term, w_(bh) can be introducedto adjust for effects caused by the borehole 201 on the measurement.With these modifications, the transformation equation becomes

${\Delta\;\hat{\sigma}} = {\frac{I\left\lbrack {S\;\Delta\;\overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\;\Delta\;\overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}{\left( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} \right).}}}}$

The sensitivity function for the transformed measurement is determinedby applying the transformation to the original sensitivity function, S.Thus,

$\hat{S} = {{{\hat{S}}^{\prime} + {i{\hat{S}}^{''}}} = {\frac{S}{I\lbrack S\rbrack} = {\frac{{S^{\prime}{I\left\lbrack S^{\prime} \right\rbrack}} + {S^{''}{I\left\lbrack S^{''} \right\rbrack}}}{{{I\lbrack S\rbrack}}^{2}} + {i{\frac{{S^{''}{I\left\lbrack S^{\prime} \right\rbrack}} + {S^{\prime}{I\left\lbrack S^{''} \right\rbrack}}}{{{I\lbrack S\rbrack}}^{2}}.}}}}}$

Note that I[Ŝ]=I[Ŝ′]=1 because I[Ŝ″]=0. The parameters for theequivalent homogeneous perturbation areΔ{circumflex over (σ)}′={circumflex over (σ)}₁−σ₀ =I[Ŝ′Δσ]−I[Ŝ″ωΔ∈]Δ{circumflex over (σ)}″=ω({circumflex over (∈)}₁−∈₀)=I[Ŝ′ωΔ∈]+I[Ŝ″Δσ].

The estimate for the conductivity perturbation, Δ{circumflex over (σ)}′suppresses sensitivity (is relatively insensitive) to the dielectricconstant perturbation, and the estimate of the dielectric constantperturbation, Δ{circumflex over (σ)}″/ω suppresses sensitivity to theconductivity perturbation. This is apparent because the coefficient ofthe suppressed variable is Ŝ″. In fact, the estimate for theconductivity perturbation Δ{circumflex over (σ)}′ is independent of thedielectric constant perturbation provided that deviations in thedielectric constant from its background are such that I[Ŝ″ωΔ∈]=0. SinceI[Ŝ″]=0, this is apparently the case if ωΔ∈ is independent of position.Likewise, the estimate for the dielectric constant perturbation given byΔ{circumflex over (σ)}″/ω is independent of the conductivityperturbation provided that deviations in the conductivity from itsbackground value are such that I[Ŝ″Δσ]=0. Since I[Ŝ″]=0, this isapparently the case if Δσ is independent of position.

Turning now to FIGS. 6 a and 6 b, illustrated are plots of thesensitivity functions Ŝ′ and Ŝ″ derived from S′ and S″ for the mediumtransmitter-receiver spacing measurement shown in FIGS. 5 a and 5 cusing the transformation

$\hat{S} = {\frac{S}{I\lbrack S\rbrack}.}$

The data in FIGS. 6 c and 6 d were derived from the data in FIGS. 5 band 5 d for the Deep T-R spacing measurement. As shown in FIGS. 6 a, 6b, 6 c and 6 d, using the transformed measurements to determine theelectrical parameters of the earth formation is a substantialimprovement over the prior art. The estimates of the medium parametersare more accurate and less susceptible to errors in the estimate of thebackground medium because the calculation of the resistivity isrelatively unaffected by the dielectric constant and the calculation ofthe dielectric constant is relatively unaffected by the resistivity. Inaddition to integrating to 0, the peak values for Ŝ″ in FIGS. 6 b and 6d are significantly less than the respective peak values for Ŝ′ in FIGS.6 a and 6 c. Both of these properties are very desirable because Ŝ″ isthe sensitivity function for the variable that is suppressed.

Realization of the Transformation

In order to realize the transformation, it is desirable to have valuesof I[S] readily accessible over the range of background mediumparameters that will be encountered. One way to achieve this is tocompute the values for I[S] and then store them in a lookup table foruse later. Of course, it is not necessary to store these data in such alookup table if it is practical to quickly calculate the values for I[S]on command when they are needed. In general, the values for I[S] can becomputed by directly; however, it can be shown that

${I\lbrack S\rbrack} = {{\frac{1}{w_{0}}\frac{\partial w}{\partial\overset{\sim}{\sigma}}}❘_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}}}$

where w₀ is the expected value for the measurement in the backgroundmedium, and the indicated derivative is calculated using the followingdefinition:

${\frac{\partial w}{\partial\overset{\sim}{\sigma}}❘_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}}} = {\lim\limits_{{\Delta\;\overset{\sim}{\sigma}}\rightarrow 0}{\frac{{w\left( {{\overset{\sim}{\sigma}}_{0} + {\Delta\;\overset{\sim}{\sigma}}} \right)} - {w\left( {\overset{\sim}{\sigma}}_{0} \right)}}{\left( {{\overset{\sim}{\sigma}}_{0} + {\Delta\;\overset{\sim}{\sigma}}} \right) - {\overset{\sim}{\sigma}}_{0}}.}}$

In the above formula, {tilde over (σ)}₀ may vary from point to point inthe formation 215 (the background medium may be inhomogeneous), but theperturbation Δ{tilde over (σ)} is constant at all points in theformation 215. As an example of evaluating I[S] using the above formula,consider the idealized case of a homogeneous medium with a smalltransmitter coil and two receiver coils spaced a distance L₁ and L₂ fromthe transmitter. Then,

$\begin{matrix}{w_{0} = {\left( \frac{L_{1}}{L_{2}} \right)^{3}\frac{{\exp\left( {{\mathbb{i}}\; k_{0}L_{2}} \right)}\left( {1 - {{\mathbb{i}}\; k_{0}L_{2}}} \right)}{{\exp\left( {{\mathbb{i}}\; k_{0}L_{1}} \right)}\left( {1 - {{\mathbb{i}}\; k_{0}L_{1}}} \right)}}} \\{{I\lbrack S\rbrack} = {{{\frac{1}{w_{0}}\frac{\partial w}{\partial\overset{\sim}{\sigma}}}❘_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}}} = {\frac{{\mathbb{i}}\;\omega\;\mu}{2}\left( {\frac{\left( L_{2} \right)^{2}}{1 - {{\mathbb{i}}\; k_{0}L_{2}}} - \frac{\left( L_{1} \right)^{2}}{1 - {{\mathbb{i}}\; k_{0}L_{1}}}} \right)}}}\end{matrix}$

The wave number in the background medium is k₀=√{square root over(iωμ{tilde over (σ)}₀)}, the function exp(·) is the complex exponentialfunction where exp(1)≈2.71828, and the symbol μ denotes the magneticpermeability of the earth formation. The above formula for I[S] appliesto both uncompensated (FIG. 3) and to compensated (FIG. 4) measurementsbecause the background medium has reflection symmetry about the centerof the antenna array in FIG. 4.

For the purpose of this example, the above formula is used to computethe values for I[S]=I[S′]+iI[S″]. FIG. 7 illustrates an exemplary table701 employed in a Create Lookup Table step 903 (FIG. 9) of the techniqueof the disclosed embodiment. Step 903 generates a table such as table701 including values for the integral of the sensitivity function overthe range of variables of interest. The first two columns of the table701 represent the conductivity σ₀ and the dielectric constant ∈₀ of thebackground medium. The third and fourth columns of the table 701represent calculated values for the functions I[S′] and I[S″] for a Deepmeasurement, in which the spacing between the transmitter 305 receivers307 and 309 is 50 and 60 inches, respectively. The fifth and sixthcolumns of the table 701 represent calculated values for the functionsI[S′] and I[S″] for a Medium measurement, in which the spacing betweenthe transmitter 305 receivers 307 and 309 is 20 and 30 inches,respectively. It is understood that both the frequency of thetransmitter(s) and the spacing between the transmitter(s) andreceiver(s) can be varied. Based upon this disclosure, it is readilyapparent to those skilled in the art that algorithms such as the onedescribed above can be applied to alternative measurementconfigurations. If more complicated background media are used, forexample including the mandrel with finite-diameter antennas, it may bemore practical to form a large lookup table such as table 701 but withmany more values. Instead of calculating I[S] every time a value isneeded, data would be interpolated from the table. Nonetheless, table701 clearly illustrates the nature of such a lookup table. Such a tablewould contain the values of the functions I[S′] and I[S″] for the entirerange of values of the conductivity σ₀ and the dielectric constant ∈₀likely to be encountered in typical earth formations. For example, I[S′]and I[S″] could be calculated for values of ∈₀ between 1 and 1000 andfor values of σ₀ between 0.0001 and 10.0. Whether calculating values forthe entire lookup table 701 or computing the I[S′] and I[S″] on commandas needed, the data is used as explained below.

FIG. 8 illustrates a chart 801 used to implement a Determine BackgroundMedium Parameters step 905 (FIG. 9) of the techniques of the disclosedembodiment. The chart 801 represents a plot of the attenuation and phaseshift as a function of resistivity and dielectric constant in ahomogeneous medium. Similar plots can be derived for more complicatedmedia. However, the homogeneous background media are routinely used dueto their simplicity. Well known numerical methods such as inverseinterpolation can be used to calculate an initial estimate of backgroundparameters based upon the chart 801. In one embodiment, the measuredattenuation and phase shift values are averaged over a few feet of depthwithin the borehole 201. These average values are used to determine thebackground resistivity and dielectric constant based upon the chart 801.It should be understood that background medium parameters can beestimated in a variety of ways using one or more attenuation and phasemeasurements.

FIG. 9 is a flowchart of an embodiment of the disclosed transformationtechniques that can be implemented in a software program which isexecuted by a processor of a computing system such as a computer at thesurface or a “downhole” microprocessor. Starting in a Begin Analysisstep 901, control proceeds immediately to the Create Lookup Table step903 described above in conjunction with FIG. 7. In an alternativeembodiment, step 903 can be bypassed and the function of the lookuptable replaced by curve matching, or “forward modeling.” Control thenproceeds to an Acquire Measured Data Step 904. Next, control proceeds toa Determine Background Values Step 905, in which the background valuesfor the background medium are determined. Step 905 corresponds to thechart 801 (FIG. 8).

Control then proceeds to a Determine Integral Value step 907. TheDetermine Integral Value step 907 of the disclosed embodiment determinesan appropriate value for I[S] using the lookup table generated in thestep 903 described above or by directly calculating the I[S] value asdescribed in conjunction with FIG. 7. Compute Parameter Estimate, step909, computes an estimate for the conductivity and dielectric constantas described above using the following equation:

${\Delta\;\hat{\sigma}} = {\frac{I\left\lbrack {S\;\Delta\;\overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\;\Delta\overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}{\left( {\frac{{c\; w_{1}} - w_{b\; h}}{w_{0}} - 1} \right).}}}}$

where the borehole effect and a calibration factor are taken intoaccount using the factors w_(bh) and c, respectively. The conductivityvalue plotted on the log (this is the value correlated to theconductivity of the actual earth formation) is Re(Δ{circumflex over(σ)}+{tilde over (σ)}₀) where the background medium is characterized by{tilde over (σ)}₀. The estimate for the dielectric constant can also beplotted on the log (this value is correlated to the dielectric constantof the earth formation), and this value is Im(Δ{circumflex over(σ)}+{tilde over (σ)}₀)/ω. Lastly, in the Final Depth step 911, it isdetermined whether the tool 201 is at the final depth within the earthformation 215 that will be considered in the current logging pass. Ifthe answer is “Yes,” then control proceeds to a step 921 where isprocessing is complete. If the answer in step 911 is “No,” controlproceeds to a Increment Depth step 913 where the tool 220 is moved toits next position in the borehole 201 which penetrates the earthformation 215. After incrementing the depth of the tool 220, controlproceeds to step 904 where the process of steps 904, 905, 907, 909 and911 are repeated. It should be understood by those skilled in the artthat embodiments described herein in the form a computing system or as aprogrammed electrical circuit can be realized.

Improved estimates for the conductivity and/or dielectric constant canbe determined by simultaneously considering multiple measurements atmultiple depths. This procedure is described in more detail below underthe heading “Multiple Sensors At Multiple Depths.”

Multiple Sensors at Multiple Depths

In the embodiments described above, the simplifying assumption thatΔ{tilde over (σ)} is not position dependent facilitates determining avalue for Δ{circumflex over (σ)} associated with each measurement byconsidering only that measurement at a single depth within the well (atleast given a background value {tilde over (σ)}₀). It is possible toeliminate the assumption that Δ{tilde over (σ)} is independent ofposition by considering data at multiple depths, and in general, to alsoconsider multiple measurements at each depth. An embodiment of such atechnique for jointly transforming data from multiple MWD/LWD sensors atmultiple depths is given below. Such an embodiment can also be used forprocessing data from a wireline dielectric tool or a wireline inductiontool. Alternate embodiments can be developed based on the teachings ofthis disclosure by those skilled in the art.

In the disclosed example, the background medium is not assumed to be thesame at all depths within the processing window. In cases where it ispossible to assume the background medium is the same at all depthswithin the processing window, the system of equations to be solved is inthe form of a convolution. The solution to such systems of equations canbe expressed as a weighted sum of the measurements, and the weights canbe determined using standard numerical methods. Such means are known tothose skilled in the art, and are referred to as “deconvolution”techniques. It will be readily understood by those with skill in the artthat deconvolution techniques can be practiced in conjunction with thedisclosed embodiments without departing from the spirit of theinvention, but that the attendant assumptions are not necessary topractice the disclosed embodiments in general.

Devices operating at multiple frequencies are considered below, butmultifrequency operation is not necessary to practice the disclosedembodiments. Due to frequency dispersion (i.e., frequency dependence ofthe dielectric constant and/or the conductivity value), it is notnecessarily preferable to operate using multiple frequencies. Given thedisclosed embodiments, it is actually possible to determine thedielectric constant and resistivity from single-frequency data. In fact,the disclosed embodiments can be used to determine and quantifydispersion by separately processing data sets acquired at differentfrequencies. In the below discussion, it is understood that subsets ofdata from a given measurement apparatus or even from several apparatusescan be processed independently to determine parameters of interest. Thebelow disclosed embodiment is based on using all the data availablestrictly for purpose of simplifying the discussion.

Suppose multiple transmitter-receiver spacings are used and that eachtransmitter is excited using one or more frequencies. Further, supposedata are collected at multiple depths in the earth formation 215. Let Ndenote the number of independent measurements performed at each ofseveral depths, where a measurement is defined as the data acquired at aparticular frequency from a particular set of transmitters and receiversas shown in FIGS. 3 or 4. Then, at each depth z_(k), a vector of all themeasurements can be defined as

${{\overset{\_}{v}}_{k} = \left\lbrack {\left( {\frac{w_{1}}{w_{0}} - 1} \right)_{k1},\left( {\frac{w_{1}}{w_{0}} - 1} \right)_{k2},\mspace{11mu}\ldots\mspace{11mu},\left( {\frac{w_{1}}{w_{0}} - 1} \right)_{kN}} \right\rbrack^{T}},$

and the perturbation of the medium parameters from the background mediumvalues associated with these measurements isΔ{tilde over ( σ=Δ{tilde over (σ)}(ρ,z)[1,1, . . . ,1]^(T)

in which the superscript T denotes a matrix transpose, v _(k) is avector each element of which is a measurement, and Δ{tilde over ( σ is avector each element of which is a perturbation from the backgroundmedium associated with a corresponding element of v _(k) at the point P225. In the above, the dependence of the perturbation, Δ{tilde over(σ)}(ρ, z) on the position of the point P 225 is explicitly denoted bythe variables ρ and z. In general, the conductivity and dielectricconstant of both the background medium and the perturbed medium dependon ρ and z; consequently, no subscript k needs to be associated withΔ{tilde over (σ)}(ρ, z), and all elements of the vector Δ{tilde over ( σare equal. As described above, borehole corrections and a calibrationcan be applied to each measurement, but here they are omitted forsimplicity.

The vectors v _(k) and Δ{tilde over ( σ are related as follows:v _(k)=I[ SΔ{tilde over ( σ]in which S is a diagonal matrix with each diagonal element being thesensitivity function centered on the depth z_(k), for the correspondingelement of v _(k), and the integral operator I is defined by:

I[F] = ∫_(−∞)^(+∞) 𝕕z∫₀^(+∞)𝕕ρ F (ρ, z).

Using the notation

I_(m n)[F] = ∫_(z_(m − 1))^(z_(m)) 𝕕z∫_(ρ_(n − 1))^(ρ_(n))𝕕ρ F (ρ, z)

to denote integrals of a function over the indicated limits ofintegration, it is apparent that

${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}\;{\sum\limits_{n = 1}^{N^{\prime}}\;{I_{m\; n}\left\lbrack {\overset{\overset{\_}{\_}}{S}\Delta\;\overset{\overset{\_}{\sim}}{\sigma}} \right\rbrack}}}$

if ρ₀=0, ρ_(N′)=+∞, z_(−M−1)=−∞, and z_(M)=+∞. The equation directlyabove is an integral equation from which an estimate of Δ{tilde over(σ)}(ρ,z) can be calculated. With the definitions

ρ_(n)*=(ρ_(n)+ρ_(n−1))/2 and z_(m)*=(z_(m)+z_(m−1))/2 and making theapproximation Δ{tilde over (σ)}(ρ, z)=Δ{circumflex over (σ)}(ρ_(n)*,z_(m)*) within the volumes associated with each value for m and n, itfollows that

${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}\;{\sum\limits_{n = 1}^{N^{\prime}}\;{{I_{m\; n}\left\lbrack \overset{\overset{\_}{\_}}{S} \right\rbrack}\Delta\;{\overset{\overset{\_}{\hat{}}}{\sigma}\left( {\rho_{n}^{*},z_{m}^{*}} \right)}}}}$

where N′≦N to ensure this system of equations is not underdetermined.The unknown values Δ{circumflex over ( σ(ρ_(n)*, z_(m)*) can then bedetermined by solving the above set of linear equations. It is apparentthat the embodiment described in the section entitled “REALIZATION OFTHE TRANSFORMATION” is a special case of the above for which M=0,N=N′=1.

Although the approximation Δ{tilde over (σ)}(ρ, z)=Δ{circumflex over(σ)}(ρ_(n)*, z_(m)*) (which merely states that Δ{tilde over (σ)}(ρ, z)is a piecewise constant function of ρ, z) is used in the immediatelyabove embodiment, such an approximation is not necessary. Moregenerally, it is possible to expand Δ{tilde over (σ)}(ρ, z) using a setof basis functions, and to then solve the ensuing set of equations forthe coefficients of the expansion. Specifically, suppose

$\mspace{65mu}{{\Delta\;{\overset{\sim}{\sigma}\left( {\rho,z} \right)}} = {\sum\limits_{m = {- \infty}}^{\infty}\;{\sum\limits_{n = {- \infty}}^{\infty}{a_{m\; n}{\phi_{m\; n}\left( {\rho,z} \right)}}}}}$${then},\mspace{59mu}{{\overset{\_}{v}}_{k} = {\sum\limits_{m = {- \infty}}^{\infty}\;{\sum\limits_{n = {- \infty}}^{\infty}{{I\left\lbrack {\overset{\overset{\_}{\_}}{S}\phi_{m\; n}} \right\rbrack}{\overset{\_}{a}}_{m\; n}}}}}$

where α _(mn)=α_(mn)[1,1, . . . ,1]^(T). Some desirable properties forthe basis functions φ_(mn) are: 1) the integrals I[ Sφ_(mn)] in theabove equation all exist; and, 2) the system of equations for thecoefficients α_(mn) is not singular. It is helpful to select the basisfunctions so that a minimal number of terms is needed to form anaccurate approximation to Δ{tilde over (σ)}(ρ, z).

The above embodiment is a special case for which the basis functions areunit step functions. In fact, employing the expansion

${\Delta\;{\overset{\sim}{\sigma}\left( {\rho,z} \right)}} = {\sum\limits_{m = {- M}}^{+ M}{\sum\limits_{n = 1}^{N^{\prime}}{\Delta\;{{{\overset{\_}{\hat{\sigma}}\left( {\rho_{n}^{*},z_{m}^{*}} \right)}\left\lbrack {{u\left( {z - z_{m}} \right)} - {u\left( {z - z_{m - 1}} \right)}} \right\rbrack}\left\lbrack {{u\left( {\rho - \rho_{n}} \right)} - {u\left( {\rho - \rho_{n - 1}} \right)}} \right\rbrack}}}}$

where u(·) denotes the unit step function leads directly to the samesystem of equations

${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}{\sum\limits_{n = 1}^{N^{\prime}}{{I_{mn}\left\lbrack \overset{\_}{\overset{\_}{S}} \right\rbrack}\Delta\;{\overset{\_}{\hat{\sigma}}\left( {\rho_{n}^{*},z_{m}^{*}} \right)}}}}$

given in the above embodiment. Specific values for M, N′, z_(m), andρ_(n) needed to realize this embodiment of the invention depend on theexcitation frequency(ies), on the transmitter-receiver spacings that areunder consideration, and generally on the background conductivity anddielectric constant. Different values for z_(m) and ρ_(n) are generallyused for different depth intervals within the same well because thebackground medium parameters vary as a function of depth in the well.

Solving the immediately above system of equations results in estimatesof the average conductivity and dielectric constant within the volume ofthe earth formation 215 corresponding to each integral I_(mn)[ S]. In anembodiment, the Least Mean Square method is used to determine values forΔ{circumflex over ( σ(ρ_(n)*, z_(m)*) by solving the above system ofequations. Many texts on linear algebra list other techniques that mayalso be used.

Unlike other procedures previously used for processing MWD/LWD data, thetechniques of a disclosed embodiment account for dielectric effects andprovide for radial inhomogeneities in addition to bedding interfaces byconsistently treating the signal as a complex-valued function of theconductivity and the dielectric constant. This procedure producesestimates of one variable (i.e., the conductivity) are not corrupted byeffects of the other (i.e., the dielectric constant).

A series of steps, similar to those of FIG. 9, can be employed in orderto implement the embodiment for Multiple Sensors at Multiple Depths.Since the lookup table for I_(mn)[ S] needed to realize such anembodiment could be extremely large, these values are evaluated asneeded in this embodiment. This can be done in a manner analogous to themeans described in the above section “REALIZATION OF THE TRANSFORMATION”using the following formulae:

${I_{mn}\lbrack S\rbrack} = {\left. {\frac{1}{w_{0}}\frac{\partial w}{\partial{\overset{\sim}{\sigma}}_{mn}}} \middle| {}_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}}\frac{\partial w}{\partial{\overset{\sim}{\sigma}}_{mn}} \right|_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}} = {\lim\limits_{{\Delta\;{\overset{\sim}{\sigma}}_{mn}}\rightarrow 0}{\frac{{w\left( {{\overset{\sim}{\sigma}}_{0} + {\Delta\;{\overset{\sim}{\sigma}}_{mn}}} \right)} - {w\left( {\overset{\sim}{\sigma}}_{0} \right)}}{\left( {{\overset{\sim}{\sigma}}_{0} + {\Delta\;{\overset{\sim}{\sigma}}_{mn}}} \right) - {\overset{\sim}{\sigma}}_{0}}.}}}$

where {tilde over (σ)}_(mn)=σ_(mn)+iω∈_(mn) represents the conductivityand dielectric constant of the region of space over which the integralI_(mn)[S] is evaluated. In words, I_(mn)[S] can be calculated byevaluating the derivative of the measurement with respect to the mediumparameters within the volume covered by the integration. Alternatively,one could evaluate I_(mn)[S] by directly carrying out the integration asneeded. This eliminates the need to store the values in a lookup table.

While the above exemplary systems are described in the context of anMWD/LWD system, it shall be understood that a system according to thedescribed techniques can be implemented in a variety of other loggingsystems such as wireline induction or wireline dielectric measurementsystems. Further in accordance with the disclosed techniques, it shouldbe understood that phase shift and attenuation can be combined in avariety of ways to produce a component sensitive to resistivity andrelatively insensitive to dielectric constant and a component sensitiveto dielectric constant and relatively insensitive to resistivity. In theinstance of MWD/LWD resistivity measurement systems, resistivity is thevariable of primary interest; as a result, phase shift and attenuationmeasurements can be combined to produce a component sensitive toresistivity and relatively insensitive dielectric constant.

Single Measurements at a Single Depth

One useful embodiment is to correlate (or alternatively equate) a singlemeasured value w₁ to a model that predicts the value of the measurementas a function of the conductivity and dielectric constant within aprescribed region of the earth formation. The value for the dielectricconstant and conductivity that provides an acceptable correlation (oralternatively solves the equation) is then used as the final result(i.e., correlated to the parameters of the earth formation). Thisprocedure can be performed mathematically, or graphically. Plotting apoint on a chart such as FIG. 8 and then determining which dielectricvalue and conductivity correspond to it is an example of performing theprocedure graphically. It can be concluded from the preceding sections,that Ŝ is the sensitivity of such an estimate of the dielectric constantand conductivity to perturbations in either variable. Thus such aprocedure results in an estimate for the conductivity that has no netsensitivity to the changes in the dielectric constant and an estimatefor the dielectric constant that has no net sensitivity to changes inthe conductivity within the volume in question. This is a very desirableproperty for the results to have. The utility of employing a singlemeasurement at a single depth derives from the fact that data processingalgorithms using minimal data as inputs tend to provide results quicklyand reliably. This procedure is a novel means of determining oneparameter (either the conductivity or the dielectric constant) with nonet sensitivity to the other parameter. Under the old assumptions, thisprocedure would appear to not be useful for determining independentparameter estimates.

Iterative Forward Modeling and Dipping Beds

The analysis presented above has been carried out assuming a2-dimensional geometry where the volume P 225 in FIG. 2 is a solid ofrevolution about the axis of the tool. In MWD/LWD and wirelineoperations, there are many applications where such a 2-dimensionalgeometry is inappropriate. For example, the axis of the tool oftenintersects boundaries between different geological strata at an obliqueangle. Practitioners refer to the angle between the tool axis and avector normal to the strata as the relative dip angle. When the relativedip angle is not zero, the problem is no longer 2-dimensional. However,the conclusion that: 1) the attenuation measurement is sensitive to theconductivity in the same volume as the phase measurement is sensitive tothe dielectric constant; and, 2) an attenuation measurement is sensitiveto the dielectric constant in the same volume that the phase measurementis sensitive to the conductivity remains true in the more complicatedgeometry. Mathematically, this conclusion follows from theCauchy-Reimann equations which still apply in the more complicatedgeometry (see the section entitled “SENSITIVITY FUNCTIONS”). Thephysical basis for this conclusion is that the conduction currents arein quadrature (90 degrees out of phase) with the displacement currents.At any point in the formation, the conduction currents are proportionalto the conductivity and the displacement currents are proportional tothe dielectric constant.

A common technique for interpreting MWD/LWD and wireline data inenvironments with complicated geometry such as dipping beds is to employa model which computes estimates for the measurements as a function ofthe parameters of a hypothetical earth formation. Once model inputparameters have been selected that result in a reasonable correlationbetween the measured data and the model data over a given depthinterval, the model input parameters are then correlated to the actualformation parameters. This process is often referred to as “iterativeforward modeling” or as “Curve Matching,” and applying it in conjunctionwith the old assumptions, leads to errors because the volumes in whicheach measurement senses each variable have to be known in order toadjust the model parameters appropriately.

The algorithms discussed in the previous sections can also be adaptedfor application to data acquired at non-zero relative dip angles.Selecting the background medium to be a sequence of layers having theappropriate relative dip angle is one method for so doing.

Transformations for a Resistivity-Dependent Dielectric Constant

In the embodiments described above, both the dielectric constant andconductivity are treated as independent quantities and the intent is toestimate one parameter with minimal sensitivity to the other. As shownin FIG. 1, there is empirical evidence that the dielectric constant andthe conductivity can be correlated. Such empirical relationships arewidely used in MWD/LWD applications, and when they hold, one parametercan be estimated if the other parameter is known.

This patent application shows that: 1) an attenuation measurement issensitive to the conductivity in the same volume of an earth formationas the phase measurement is sensitive to the dielectric constant; and,2) the attenuation measurement is sensitive to the dielectric constantin the same volume that the phase measurement is sensitive to theconductivity. A consequence of these relationships is that it is notgenerally possible to derive independent estimates of the conductivityfrom a phase and an attenuation measurement even if the dielectricconstant is assumed to vary in a prescribed manner as a function of theconductivity. The phrase “not generally possible” is used above becauseindependent estimates from each measurement can be still be made if thedielectric constant doesn't depend on the conductivity or if theconductivity and dielectric constant of earth formation are practicallythe same at all points within the sensitive volumes of bothmeasurements. Such conditions represent special cases which are notrepresentative of conditions typically observed within earth formations.

Even though two independent estimates of the conductivity are notgenerally possible from a single phase and a single attenuationmeasurement, it is still possible to derive two estimates of theconductivity from a phase and an attenuation measurement given atransformation to convert the dielectric constant into a variable thatdepends on the resistivity. For simplicity, consider a device such asthat of FIG. 3. Let the complex number w₁ denote an actual measurement(i.e., the ratio of the voltage at receiver 307 relative to the voltageat receiver 309, both voltages induced by current flowing throughtransmitter 305). Let the complex number w denote the value of saidmeasurement predicted by a model of the tool 320 in a prescribed earthformation 315. For further simplicity, suppose the model is as describedabove in the section “REALIZATION OF THE TRANSFORMATION.” Then,

${w \equiv {w\left( {\sigma,{ɛ(\sigma)}} \right)}} = {\left( \frac{L_{1}}{L_{2}} \right)^{3}\frac{{\exp\left( {ikL}_{2} \right)}\left( {1 - {ikL}_{2}} \right)}{{\exp\left( {ikL}_{1} \right)}\left( {1 - {ikL}_{1}} \right)}}$

where the wave number k≡k(σ, ∈(σ))=√{square root over (iωμ(σ+iω∈(σ)))},and the dependence of the dielectric constant ∈ on the conductivity σ isaccounted for by the function ∈(σ). Different functions ∈(σ) can beselected for different types of rock. Let σ_(P) and σ_(A) denote twoestimates of the conductivity based on a phase and an attenuationmeasurement and a model such as the above model. The estimates can bedetermined by solving the system of equations0=|w ₁ |−|w(σ_(A),∈(σ_(P))) |0=phase(w ₁)−phase(w(σ_(P),∈(σ_(A)))).

The first equation involves the magnitude (a.k.a. the attenuation) ofthe measurement and the second equation involves the phase (a.k.a. thephase shift) of the measurement. Note that the dielectric constant ofone equation is evaluated using the conductivity of the other equation.This disclosed technique does not make use of the “old assumptions.”Instead, the attenuation conductivity is evaluated using a dielectricvalue consistent with the phase conductivity and the phase conductivityis evaluated using a dielectric constant consistent with the attenuationconductivity. These conductivity estimates are not independent becausethe equations immediately above are coupled (i.e., both variables appearin both equations). The above described techniques represent asubstantial improvement in estimating two resistivity values from aphase and an attenuation measurement given a priori information aboutthe dependence of the dielectric constant on the conductivity. It can beshown that the sensitivity functions for the conductivity estimatesσ_(A) and σ_(P) are S′ and S″, respectively if the perturbation to thevolume P 225 is consistent with the assumed dependence of the dielectricconstant on the conductivity and σ_(A)=σ_(P).

It will be evident to those skilled in the art that a more complicatedmodel can be used in place of the simplifying assumptions. Such a modelmay include finite antennas, metal or insulating mandrels, formationinhomogeneities and the like. In addition, other systems of equationscould be defined such as ones involving the real and imaginary parts ofthe measurements and model values. As in previous sections of thisdisclosure, calibration factors and borehole corrections may be appliedto the raw data.

Transformations for a Resistivity-Dependent Dielectric Constant Used inConjunction with Conventional Phase Resistivity Values

The preceding section, “TRANSFORMATIONS FOR A RESISTIVITY-DEPENDENTDIELECTRIC CONSTANT,” provides equations for σ_(P) and σ_(A) that do notmake use of the “old assumptions.” One complication that results fromusing said equations is that both a phase shift measurement and anattenuation measurement must be available in order to evaluate eitherσ_(P) or σ_(A) In MWD/LWD applications, both phase shift and attenuationmeasurements are commonly recorded; however, the attenuationmeasurements are often not telemetered to the surface while drilling.Since bandwidth in the telemetry system associated with the logging toolis limited, situations arise where it is useful to have less preciseresistivity measurements in favor of other data such as density, speedof sound or directional data. The following reparameterization of thesystem of equations in the previous section accommodates this additionalconsideration:0=|w ₁ |−|w(σ_(A),∈(σ_(P)))|0=phase(w ₁)−phase(w(σ_(P),∈(σ_(P)))),

where σ_(P) and σ_(A) respectively represent the reciprocals of thefirst and second resistivity values; the second equation involves aphase of the measured electrical signal and the first equation involvesa magnitude of the measured electrical signal at a given frequency ofexcitation; a function ∈(·) represents a correlation between thedielectric constant and the resistivity and is evaluated in σ_(p) in thefirst and second equations; w₁ represents an actual measurement (e.g.,the ratio of the voltage at receiver 307 relative to the voltage atreceiver 309, both voltages induced by current flowing throughtransmitter 305) in the form of a complex number, and a function w(σ,∈)represents a mathematical model which estimates w₁. More particularly,∈(σ_(P)) represents the transformation of dielectric constant into avariable that depends upon resistivity.

The second equation evaluates phase conductivity σ_(P) and thedielectric constant correlation with the same phase conductivity whichis consistent with sensing both resistivity and dielectric constant insubstantially the same volume. The first equation evaluates attenuationconductivity σ_(A) and the dielectric constant correlation with thephase conductivity (not an attenuation conductivity) which is consistentwith sensing the resistivity and dielectric constant in differentvolumes. The second equation allows σ_(P) to be determined by a phaseshift measurement alone. The first equation is actually the sameequation as in the previous section, but the value of σ_(A) that solvesthe equation may not be the same as in the previous section because thevalue of σ_(P) used in the correlation between the dielectric constantand the resistivity may not be the same. In other words, the phaseconductivity σ_(P) can be determined without the attenuationmeasurement, but the attenuation conductivity σ_(A) is a function ofboth attenuation and phase shift measurements. The value of σ_(P)resulting from these equations is the conventional phase conductivitycited in the prior art which is derived using the “old assumptions” thatthe phase shift measurement senses both the resistivity and dielectricconstant within substantially the same volume. The value for σ_(A) isconsistent with the fact the attenuation measurement senses theresistivity in substantially the same volume that the phase measurementsenses the dielectric constant; however, the σ_(A) value is a betterapproximation than the σ_(A) values in the previous section because thephase conductivity σ_(P) used to evaluate the correlation between thedielectric constant and the resistivity is derived using the “oldassumptions.” The above two equations are partially coupled in that thephase conductivity σ_(P) is needed from the second equation to determinethe attenuation conductivity σ_(A) in the first equation.

Since the phase shift measurement is typically less sensitive to thedielectric constant than the corresponding attenuation measurement, theequations given in this section for σ_(P) and σ_(A) provide a reasonabletradeoff between the following: 1) accurate resistivity measurementresults, 2) consistent resistivity measurement results (reporting thesame resistivity values from both recorded and telemetered data), and 3)minimizing amounts of telemetered data while drilling.

When the perturbation to the volume P 225 is consistent with the assumeddependence of the dielectric constant on the conductivity, thesensitivity functions for conductivity estimates σ_(P) and σ_(A)determined by the equations given in this section are:

${\Delta\;\sigma_{P}} = {\frac{S^{''}\omega\frac{\mathbb{d}ɛ}{\mathbb{d}\sigma}S^{\prime}}{{I\left\lbrack S^{''} \right\rbrack} - {\omega\frac{\mathbb{d}ɛ}{\mathbb{d}\sigma}{I\left\lbrack S^{\prime} \right\rbrack}}}\Delta\;\sigma\;{\Delta\rho\Delta}\; z}$${\Delta\;\sigma_{A}} = {\left\lbrack {\frac{S^{\prime}}{I\left\lbrack S^{\prime} \right\rbrack} + \frac{\left( {\omega\frac{\mathbb{d}ɛ}{\mathbb{d}\sigma}} \right)^{2}\left( {{S^{\prime}{I\left\lbrack S^{''} \right\rbrack}} - {S^{''}{I\left\lbrack S^{\prime} \right\rbrack}}} \right)}{{I\left\lbrack S^{\prime} \right\rbrack}\left( {{I\left\lbrack S^{''} \right\rbrack} - {\omega\frac{\mathbb{d}ɛ}{\mathbb{d}\sigma}{I\left\lbrack S^{\prime} \right\rbrack}}} \right)}} \right\rbrack\Delta\;{\sigma\Delta}\;{\rho\Delta}\; z}$

where

$\frac{\mathbb{d}ɛ}{\mathbb{d}\sigma}$represents the derivative of function ∈(σ), and it was assumed forsimplicity that σ_(A)=σ_(P). The sensitivity function for σ_(P) is thecoefficient of ΔσΔρΔz in the first equation, and the sensitivityfunction for σ_(A) is the coefficient of ΔσΔρΔz in the second equation.Note that the leading term S″ in the sensitivity function of the firstequation and the leading term

$\frac{S^{\prime}}{I\left\lbrack S^{\prime} \right\rbrack}$in the sensitivity function of the second equation are approximately thesensitivity function mentioned in the section, “TRANSFORMATIONS FOR ARESISTIVITY-DEPENDENT DIELECTRIC CONSTANT,” and that the remaining termsare representative of the sensitivity error resulting from using theequations given in this section. In the second equation, the error termin the sensitivity for σ_(A) tends to be small because it isproportional to

$\left( {\omega\frac{\mathbb{d}ɛ}{\mathbb{d}\sigma}} \right)^{2}.$It can be shown that the error term for the sensitivity function of aσ_(A) estimate resulting from equations consistent with the oldassumptions is proportional to

$\left( {\omega\frac{\mathbb{d}ɛ}{\mathbb{d}\sigma}} \right)$and is thus a larger error than produced by the technique described inthis section. As a result, the technique of this section represents asubstantial improvement over the prior art.Approximations to Facilitate Use of Rapidly-Evaluated Models

Many of the techniques described in the previous sections are even moreuseful if they can be applied in conjunction with a simplified model ofthe measurement device. This is especially true for embodiments whichuse iterative numerical techniques to solve systems of nonlinearequations because the amount of computer time required to achieve asolution is reduced if details of the measurement device can be ignored.Such details include finite sized antennas 205, 207 and a metallic drillcollar 203. Models that do not include such details can often beevaluated rapidly in terms of algebraic functions whereas modelsincluding these details may require a numerical integration or othersimilar numerical operations. An alternative choice for solving suchsystems of equations is to store multidimensional lookup tables andperform inverse interpolation. An advantage of using the lookup tablesis that the model calculations are done in advance; so, results can bedetermined quickly once the table has been generated. However, 1) alarge amount of memory may be required to store the lookup table; 2) itmay be difficult to handle inputs that are outside the range of thetable; 3) the tables may require regeneration if the equipment ismodified, and 4) the tables themselves may be costly to generate andmaintain.

The approach taken here is a compromise. One-dimensional lookup tablesare used to renormalize each measurement so as to be approximatelyconsistent with data from a simplified measurement device which hasinfinitesimal antennas and no metallic mandrel. The renormalized dataare used in conjunction with a simpler, but more rapidly evaluated modelto determine the conductivity and/or dielectric constant estimates.Higher dimensional lookup tables could be used to account for morevariables, but this has proved unnecessary in practice.

Suppose the function h₀(σ,∈) represents a model that estimates themeasurements as a function of the conductivity and dielectric constantwhich includes details of the tool that are to be normalized away (i.e.finite antennas and a metallic mandrel). Suppose h₁(σ,∈) is a simplifiedmodel which is a function of the same formation parameters (i.e. theresistivity and the dielectric constant) but that assumes infinitesimalantennas and no metallic mandrel. The data shown in FIG. 10 are for themedium spaced 2 MHz measurement described in conjunction with FIG. 7.The first two columns of data resistivity (1/sigma) and dielectricconstant (eps_rel) values are input into the respective models (whichare not used in the algorithm, but are shown to illustrate how the tableis generated). It was found that when the resistivity and dielectricconstant are not both large, satisfactory results can be achieved bycalculating the data as a function of the conductivity for only onedielectric constant, and in this embodiment, a relative dielectricconstant of 35 was used. The third column (db_pt) contains theattenuation values for the simplified model evaluated as a function ofconductivity, and the fourth column (db_man) contains the attenuationvalues for the model which explicitly accounts for the finite sizedantennas and metallic mandrel. The fifth (deg_pt) and sixth (deg_man)columns are similar to the third and fourth columns but for the phaseshift instead of the attenuation. All data are calibrated to read zeroif the electrical parameters of the surrounding medium are that of avacuum (i.e., the air) (σ=0,∈=1). Specifically, columns 3 and 4 are:g′ _(a) =db _(—) pt=20 log 10(|h ₁(σ,∈)/h ₁(0,1)|)g _(a) =db_man=20 log 10(|h ₀(σ,∈)/h ₀(0,1)|)

Columns 5 and 6 are:

$g_{p}^{\prime} = {{deg\_ pt} = {\frac{180}{\pi}{\arg\left( {{h_{1}\left( {\sigma,ɛ} \right)}/{h_{1}\left( {0,1} \right)}} \right)}}}$$g_{p} = {{deg\_ man} = {\frac{180}{\pi}{{\arg\left( {{h_{0}\left( {\sigma,ɛ} \right)}/{h_{0}\left( {0,1} \right)}} \right)}.}}}$

In practice, calibrated measurements are used as values for db_man orphs_man (i.e. the ordinates in the one-dimensional interpolation). Thecorresponding values for db_pt or phs_pt result from the interpolation.Once the values for db_pt or phs_pt have been determined, relativelysimple models such as

$h_{1} = {\left( \frac{L_{1}}{L_{2}} \right)^{3}\frac{{\exp\left( {{\mathbb{i}}\; k_{0}L_{2}} \right)}\left( {1 - {{\mathbb{i}}\; k_{0}L_{2}}} \right)}{{\exp\left( {{\mathbb{i}}\; k_{0}L_{1}} \right)}\left( {1 - {{\mathbb{i}}\; k_{0}L_{1}}} \right)}}$

can be parameterized as described in the several preceding sections andused to solve numerically for the desired parameters. The solutions canbe obtained quickly, using robust and well known numerical means such asthe nonlinear least squares technique.

An alternative procedure would be to transform the values from therelatively simple model, to corresponding db_man and deg_man values.This would produce equivalent results, but in conjunction with aniterative solution method, is clumsy because it requires the values fromthe simple model to be converted to db_man and deg_man values at eachstep in the iteration.

Extended Approximations to Facilitate Use of Rapidly-Evaluated Models

The embodiment described in above section, “APPROXIMATIONS TO FACILITATEUSE OF RAPIDLY-EVALUATED MODELS,” looses accuracy when the resistivityand dielectric constant are simultaneously large enough to cause theattenuation value to be less than its “air-hang” value. This regioncorresponds to values in FIG. 8 with negative values on the horizontalaxis. Embodiments of an extended technique addressing this shortcomingare described below. This extension is based on the observation that thedifference between the mandrel and point-dipole values depends morestrongly on the dielectric constant than the resistivity when theresistivity is large.

When the attenuation and phase shift measurements are consistent withsimultaneous large resistivity and dielectric constant values, the datain FIG. 11 are used (instead of the data in FIG. 10) to transform themeasured values (deg_man, db_man) to the corresponding point-dipolevalues (deg_pt, db_pt). FIG. 12 is a plot of the difference between thepoint-dipole and mandrel values as a function of both the resistivityand dielectric constant. Two regimes are evident: 1) for lowresistivity, the difference between the point-dipole and mandrel datadepends more strongly on the resistivity than on the dielectricconstant; and, 2) for high resistivity, the difference between thepoint-dipole and mandrel data depends more strongly on the dielectricconstant than on the resistivity.

In order to facilitate use of these extended approximations withoutoperator intervention, point-dipole values consistent with both FIGS. 10and 11 are computed. They are combined to form a weighted sum thatbiases the result heavily toward consistency with FIG. 11 whenappropriate and heavily toward consistency with FIG. 10 whenappropriate. Specifically, for the disclosed embodiment two terms areused in the weighted sum,g′ _(a) =c _(1a)ƒ_(1a)(g_(a))+c _(2a)ƒ_(2a)(g _(a))g′ _(p) =c _(1p)ƒ_(1p)(g_(p))+c _(2p)ƒ_(2p)(g _(p))

where g′_(a) and g′_(p) represent the point-dipole values, g_(a) andg_(p) represent the measured (mandrel) values as defined under theheading, “APPROXIMATIONS TO FACILITATE USE OF RAPIDLY-EVALUATED MODELS.”The functions ƒ_(1a) and ƒ_(1p), defined numerically from data such asthat of FIG. 10, give corresponding point-dipole value as a function ofthe measured value for a first set of resistivity and dielectricconstant values. The functions ƒ_(2a) and ƒ_(2p), defined numericallyfrom data such as that of FIG. 11, give corresponding point-dipole valueas a function of the measured value for a second set of resistivity anddielectric constant values. The coefficients c_(1a), c_(2a), c_(1p), andc_(2p) are chosen to produce values for g′_(a) and g′_(p) thatapproximate point-dipole values over the anticipated range of g_(a) andg_(p). Satisfactory performance of the algorithm has been obtained with

$\begin{matrix}{c = {c_{1a} = {c_{1p} = {{1 - c_{2a}} = {1 - c_{2p}}}}}} \\{= \frac{{u\left( {g_{a} - g_{a0}} \right)} + {{\exp\left( {{- \alpha}{{g_{a} - g_{a0}}}} \right)}{u\left( {g_{a0} - g_{a}} \right)}}}{1 + {\exp\left( {{- \alpha}{{g_{a} - g_{a0}}}} \right)}}}\end{matrix}$ where ${u(x)} = \left\{ \begin{matrix}{0,} & {x < 0} \\{0.5,} & {x = 0} \\1 & {x > 0}\end{matrix} \right.$α=5×10⁻⁵f, f is the frequency in Hz

andg _(a0)=β1n(g _(p))+γ

For each spacing and frequency the parameters β and γ can be determinedempirically. For the medium 2 MHz measurement discussed above, β=−0.047dB and γ=−0.159 dB are satisfactory.

FIG. 13 shows the results of applying this embodiment to the medium 2MHz data. The dotted lines represent the measured (mandrel) data g_(a)and g_(p). The dot-dash lines represent the corrected data g′_(a) andg′_(p). Ideally, the corrected data would coincide exactly with theactual point-dipole values which the solid lines represent. Thedifference between such ideal results and the results achieved usingthis embodiment are not significant for most practical applications.

It will be clear to those skilled in the art that further improvementsin accuracy can be achieved if necessary. One way would be to useindependent values for the coefficients c_(1a) c_(2a), c_(1p), andc_(2p). Another alternative would be to add more terms to the weightedsums. If N terms are used,

$\begin{matrix}{g_{a}^{\prime} = {\sum\limits_{i = 1}^{N}{c_{ia}{f_{ia}\left( g_{a} \right)}}}} \\{g_{p}^{\prime} = {\sum\limits_{i = 1}^{N}{c_{ip}{f_{ip}\left( g_{p} \right)}}}}\end{matrix}$

where f_(ia) and f_(ip) represent the transformation between the mandreland point-dipole data over prescribed trajectories in the g_(a), g_(p)plane. The embodiments described above in detail employs one trajectorydetermined by constant values of resistivity as in FIG. 10 and anothertrajectory determined by constant dielectric constant values as in FIG.11. However, trajectories where neither variable is constant are alsoacceptable.

Mathematical Formulation for Iterative Forward Modeling Using the FullParameter Space

In the above section entitled, “ITERATIVE FORWARD MODELING AND DIPPINGBEDS,” dielectric constant and resistivity values are simultaneouslydetermined using procedures known as “iterative forward modeling” or“curve matching.” Those of ordinary skill in the art are generallyfamiliar with a mathematical formulation for these procedures. Aformulation is given here in order to define some notation that will beused in the next section. For simplicity, the electrical properties willbe treated as piecewise constant functions of space. Consistent with theabove text, the conductivity and dielectric values are treated asreal-valued variables. This simplifying assumption results in dielectriclosses being associated with the corresponding conductivity and in theassociation of conduction currents in quadrature with the electric fieldwith the corresponding dielectric constant.

Let the vectorw ₁ = w ₁ ′+i w ₁ ″=[w ₁₁ , . . . , w _(1N)]^(T)

denote a set of measured data where the superscript T denotes the matrixtranspose. Each component of the vector w ₁ is a complex number thatrepresents a voltage ratio defined as explained in the descriptions ofFIGS. 3 and 4. For convenience, it will be assumed that the number ofmeasurements N=JK is the product of: 1) the number of depth values Kbeing considered, and 2) the number of measurements J available at eachdepth. Suppose a model is available to estimate w ₁ as a function of thevectorsσ=[σ₁, . . . , σ_(L)]^(T)∈=[∈₁, . . . , ∈_(L)]^(T)andb=[b₁, . . . , b_(M)]^(T)

Each value σ_(i) and ∈_(i) respectively represents a conductivity valueand dielectric constant value within a prescribed region of space. Eachvalue b_(i) represents a spatial coordinate of a boundary separating twoor more regions. Specific values of each vector σ, ∈, and b to becorrelated with the parameters of the earth formation denoted by σ*, ∈*,and b* can be determined by solving the equationw ₁ = F ( σ*, ∈*, b *)

where F represents a model used to estimate the measurement vector as afunction of the vectors σ, ∈, and b. The model F is itself atransformation (or function) that maps the vectors σ, ∈, and b to avector representative of the measurement. It is desirable for theJacobian matrix of the transformation F to have a sufficient number ofsingular values within a domain near the point σ*, ∈*, and b* in orderto ensure the problem is not underdetermined. Many published numericaltechniques for obtaining approximate solutions when a system ofequations is fully, over, or under determined are known to those skilledin the art. Those of ordinary skill in the art are generally familiarwith mathematical derivation and numerical implementation ofmathematical models suitable for use as the vector function F.

In some instances, it may be helpful to apply transformations to themeasured and model data, and then solve the transformed system ofequations. The formula G( w ₁)= H( F( σ*, ∈*, b*)) represents suchtransformed equations. The system of equations w ₁= F( σ*, ∈*, b*) isjust the special case where the transformations G and H appliedrespectively to the measured and model data are both identitytransformations. Examples where transformations other than the identitytransformation are applied to the measured and model data are discussedbelow under the headings, “MATHEMATICAL FORMULATION FOR ITERATIVEFORWARD MODELING USING A REDUCED PARAMETER SPACE,” and “USE OF PHASEAND/OR ATTENUATION RESISTIVITY DATA.”

Mathematical Formulation for Iterative Forward Modeling Using a ReducedParameter Space

In the previous section, the conductivity values are solved for alongwith the corresponding dielectric constants. Solving the system ofequationsw ₁ = F ( σ*, ∈*, b *)

may involve repeated evaluation of the Jacobian matrix for thetransformation F which will be denoted by J( F). This matrix containsthe derivative of F with respect to each variable. After eachevaluation, a singular value decomposition may be performed on J( F).Evaluating and manipulating this matrix can be numerically intensive andtime consuming. Use of the Cauchy-Reimann equations to relatederivatives of F_(i) with respect to σ_(j) and ∈_(j) helps reduce thecost of evaluating J( F), but does not reduce the dimensionality of thisN×(2L+M) complex-valued matrix where N represents the number ofmeasurements, L represents the number of conductivity and dielectricconstant values and M represents the number of spatial coordinates beingtreated as variables.

In many applications, the conductivity is the value of primary interest;consequently, the disclosed embodiment is a technique of determiningvalues for σ* and b* without also determining ∈*, providing numericalefficiency. It should be evident that analogous procedures could beapplied to estimate ∈* and b* without also determining σ*. The steps ofthe disclosed embodiment are as follows:

-   -   1. Apply a first transformation to the measured data such that        the result, σ ₀, is relatively sensitive to the conductivity and        relatively insensitive to the dielectric constant: σ ₀=Re( F ₀        ⁻¹( w ₁)).    -   2. Transform the unknown dielectric constant vector ∈ which F        depends on to a vector that is a function of the corresponding        (but also unknown) conductivity vector σ. In other words, ∈→ ∈(        σ)=[∈₁(σ₁), . . . , ∈_(L)(σ_(L))]^(T). For convenience, a given        dielectric value ∈_(i) is assumed to be a function only of the        corresponding conductivity value σ_(i). Alternatively, some or        all of the dielectric values could be assumed to be: 1)        constants, independent of the conductivity; or 2) functions of        any or all of the conductivity values associated with the vector        σ.    -   3. Evaluate the model F( σ, ∈( σ), b)    -   4. Apply the transformation in step 1 to F( σ, ∈( σ), b)    -   5. Identify values for σ and b that cause the results of Steps 1        and 4 to agree, and correlate these values with σ* and b*.

A suitable choice for the transformation mentioned Step 1 is to useconductivity values determined as described above in “SINGLEMEASUREMENTS AT A SINGLE DEPTH,” and described in more detail in, S. M.Haugland, “New Discovery with Important Implications for LWD PropagationResistivity Processing and Interpretation,” SPWLA 42^(nd) Annual LoggingSymposium, paper LL, Jun. 17–20, 2001. In the embodiment disclosed here,Step 1 is realized by fitting each measurement (each component of themeasurement vector w ₁), to a model of the measurement apparatus in ahomogeneous medium and associating the conductivity of this medium withthe result of the transformation. In terms of the above notation,w ₁ = F ₀( I σ ₀ +i Ω ∈ ₀)

where the N×N matrix Ω is diagonal with each nonzero element being equalto the radian frequency of excitation of the corresponding measurement;I is the identity matrix; F ₀ is the model of the measurement apparatusin the hypothetical homogeneous medium; the vector of length N, σ ₀, isthe result of Step 1; and the vector of length N, ∈ ₀, is the dielectricconstant vector associated with σ ₀. No vector b was associated with F ₀because the medium is assumed to be homogeneous (however, aninhomogeneous background medium could be assumed without departing fromthe scope or spirit of this invention). In this embodiment, the systemof equations w ₁= F ₀( I σ ₀+i Ω ∈ ₀) is uncoupled and can be expressedin scalar notation as follows:w _(1p) =F _(0p)(σ_(0p) +iω _(p)∈_(0p)); p=1, . . . , N

The remaining steps are achieved by solving the system of equations:Re( F ₀ ⁻¹( F ( σ*, ∈( σ*), b*)))= σ ₀

where Re(·) represents the real part of its argument. The inversefunction F ₀ ⁻¹ has the properties: w ₁= F ₀( F ₀ ⁻¹( w ₁)) and σ ₀+i Ω∈ ₀= F ₀ ⁻¹( F ₀( I σ ₀+i Ω ∈ ₀)). Realizations of the inverse functionF ₀ ⁻¹ applied to the model F and measured data w ₁ may be slightlydifferent in order to account for differences between the measurementtool used to collect the measured data and the properties of themeasurement tool assumed in the model. Alternatively, the model F may bedefined so as to directly account for these differences.

Invasion Example

Instances where borehole fluids penetrate (invade) into the formationover time after drilling occur commonly. In some cases, the dataacquired during the actual drilling process reflect invasion. In othercases, resistivity measurements from MAD (measurement after drilling)passes may reflect invasion even if the data acquired during drilling(the LWD data) did not. Invaded zones are often modeled by a piston-likecylinder surrounding the wellbore. Realizations of the vectors σ, ∈, andb to characterize the invaded region of the formation are: σ=[σ_(xo),σ_(t)]^(T), ∈=[∈_(xo), ∈_(t)]^(T), and b=[r_(i)]^(T) where thesubscripts xo and t respectively denote values associated with theinvaded zone and virgin formation, and the radius of the invaded zone isr_(i).

Given a set of measured data, one could solve directly for the twoconductivity values, the two dielectric constants, and the invaded zoneradius using the above mathematical formulation for iterative forwardmodeling using the full parameter space.

In the above mathematical formulation for iterative forward modelingusing a reduced parameter space, the measured data are transformed to anapparent conductivity vector σ ₀. The reduced sensitivity of σ ₀ to thedielectric constant is a motive transforming the unknown dielectricconstant vector to a function of the unknown conductivity values: ∈→ ∈(σ)=[∈_(xo)(σ_(xo)), ∈_(t)(σ_(t))]^(T). The remaining variables σ_(xo),σ_(t) and r_(i) are then estimated by selecting values for σ and b suchthat computed estimates of the apparent conductivity values agreereasonably well with the apparent conductivity vector derived directlyfrom the measurements.

Frequency Dispersion

The section entitled “MULTIPLE SENSORS AT MULTIPLE DEPTHS,” mentionsfrequency dispersion is a complication that can arise if multipleexcitation frequencies are used. One way to deal with frequencydispersion is to process data acquired using different excitationfrequencies separately. For example, if several excitation frequenciesare used, then distinct measurement vectors would be formed for eachexcitation frequency, and distinct realizations of the vectors σ*, ∈*,and b* would be computed for each excitation frequency. Separateprocessing of multi-frequency data can be done in either the full orreduced parameter spaces.

Over the frequency range of interest in wireline induction and LWDpropagation resistivity logging applications, frequency dispersion ismore commonly observed on the dielectric constant than the conductivity.As a result, the technique for estimating the conductivity values in thereduced parameter space is expected to be relatively insensitive todielectric frequency dispersion even if multi-frequency data aresimultaneously processed and no special measures are taken to accountfor (or suppress) dielectric dispersion effects. A variation of theembodiment disclosed above that is useful for further reducing thesensitivity of the results to dielectric frequency dispersion is toapply different realizations of function ∈( σ) for each excitationfrequency.

A variation of the above mathematical formulation for iterative forwardmodeling using the full parameter space that is useful to reduce thesensitivity of those results to dielectric frequency dispersion is toassociate a different dielectric constant vector with each excitationfrequency. Doing this results in the dielectric vector becoming the L×Rmatrix

$\overset{\overset{\_}{\_}}{ɛ} = \begin{bmatrix}ɛ_{11} & \cdots & ɛ_{1R} \\\vdots & ⋰ & \vdots \\ɛ_{L1} & \cdots & ɛ_{LR}\end{bmatrix}$

where R represents the number of excitation frequencies. The unknownconductivity vector could also be formulated in terms of a matrix toaccount for conductivity frequency dispersion. Such a matrix may berepresented by:

$\overset{\overset{\_}{\_}}{\sigma} = \begin{bmatrix}\sigma_{11} & \cdots & \sigma_{1R} \\\vdots & ⋰ & \vdots \\\sigma_{L1} & \cdots & \sigma_{LR}\end{bmatrix}$

In fact, mathematical formulation for iterative forward modeling usingthe full parameter spacing can be rewritten assuming frequencydispersion on both variables by defining the variables in terms ofappropriate matrices. In that case, the equation w ₁= F( σ*, ∈*, b*)above becomesw ₁ = F ( σ*, ∈*, b *)

and situations where frequency dispersion is not accounted for can behandled by augmenting this system of equations with constraints causingsome or all of the columns of either matrix σ* or ∈* to be equal. Suchconstraint equations can be written in the form σ_(pq)=σ_(pr) or∈_(pq)=∈_(pr). Defining unknown conductivity and dielectric matrices inthe mathematical formulation for iterative modeling using a reducedparameter space to explicitly account for frequency dispersion resultsin the system of equations:Re( F ₀ ⁻¹( F ( σ*, ∈( σ*), b*)))= σ ₀

which can also be augmented by constraints of the form σ_(pq)=σ_(pr).These techniques reduce sensitivity to frequency dispersion and allowone to simultaneously process measurements from different excitationfrequencies in the presence of frequency dispersion.

Dropping the Piecewise Constant Electrical Parameter Assumption

Use of the vector b as defined in the mathematical formulation foriterative modeling using the full parameter space and thereafter is anassumption that the electrical parameters are piecewise constantfunctions of space within the earth formation. Such simplifyingassumptions are commonly applied in the art for convenience, however,more faithful representations of spatial variations in the electricalproperties are sometimes required. For example, an abrupt transitionbetween the virgin formation and invaded zone is not anticipated bymultiphase fluid flow models used to predict fluid profiles around thewellbore. Nonetheless, measured data are often fit to electromagneticmodels which assume step boundaries between adjacent regions of theearth formation. The practice seems to be common because: 1)electromagnetic models which explicitly assume continuously varying (asa function of spatial variables) electrical parameters are not readilyavailable; and 2) the spatial dependence of the resistivity and/ordielectric constant is typically not known a priori. It will be evidentto those skilled in the art that such a simplifying assumption isoptional.

Making a Dielectric Assumption without Applying a Transformation to theMeasured Data

In the mathematical formulation for iterative modeling using a reducedparameter space, the dielectric constants in the model F are assumed tobe functions of the conductivities, and the remaining unknowns areestimated to give good agreement between the apparent conductivityvalues derived from the measured data and the apparent conductivityvalues derived from data calculated using the model F. For a measurementvector w ₁ of length N, this method can be used to estimate no more thanN conductivity values. Up to 2N conductivity estimates can be derivedfrom the same measured data set if the apparent conductivity values arenot used and the model is fit directly to the measurement vector. If theassumed dielectric function ∈( σ) is a sufficiently accuraterepresentation of the actual dielectric properties fitting the modeldata directly to the measured data subject to the dielectric assumptionwill give accurate results. This embodiment consists of solving thesystem of equationsw ₁ = F ( σ*, ∈( σ*), b*)

where the constraint equations mentioned above under the heading“FREQUENCY DISPERSION” can be applied to the extent that frequencydispersion need not be accounted for.

Use of Phase and/or Attenuation Resistivity Data

Calculation of attenuation and phase resistivity values from individualmeasurements is discussed above under the headings, “TRANSFORMATIONS FORA RESISTIVITY DEPENDENT DIELECTRIC CONSTANT,” and “TRANSFORMATIONS FOR ARESISTIVITY DEPENDENT DIELECTRIC CONSTANT USED IN CONJUNCTION WITHCONVENTIONAL PHASE RESISTIVITY VALUES.” Matching these resistivityvalues to similarly calculated resistivity values derived from modeldata is another embodiment that can be used. Suppose A_(0p) denotes atransformation that converts the measured voltage ratio value w_(1p) toa corresponding attenuation resistivity and P_(0p) denotes atransformation that converts measured voltage ratio value w_(1p) to thecorresponding phase resistivity. Then, the real-valued vectors of lengthN, P ₀( w ₁) and Ā₀( w ₁) are respectively the phase and attenuationresistivity values. It is then possible to set up systems of equationsof the formP ₀( w ₁)= P ₀( F ( σ*, ∈( σ*), b*))andĀ ₀( w ₁)=Ā ₀( F ( σ*, ∈( σ*), b*))

which can either be solved simultaneously or independently, depending onthe situation. As discussed above, different versions of thetransformations P ₀ and Ā₀ can be used on the right and left hand sidesof the above equations if differences between the actual measurementtool and the measurement tool assumed in the model have not beenaccounted for by other means. It should be understood that any of theequations presented herein may be implemented in a software programexecuted by a processor of a computing system such as a computer at thesurface or a “downhole” microprocessor.

The disclosed techniques provide accurate conductivity estimates thatare independent of dielectric constant properties. Conductivity valuesserve several applications such as (i) detecting the presence, absenceor amount of a hydrocarbon, (ii) guiding a drill bit within a productivezone, (iii) estimating poor pressure, and (iv) evaluating reservoir andother geological features through correlation with logs and nearbywells. Two examples of applications for dielectric values are detectingthe presence, absence or amount of a hydrocarbon and detecting verticalfractures.

The foregoing disclosure and description of the various embodiments areillustrative and explanatory thereof, and various changes in thedescriptions, modeling, parameters and attributes of the system, theorganization of the measurements, transmitter and receiverconfigurations, and the order and timing of steps taken, as well as inthe details of the illustrated system may be made without departing fromthe spirit of the invention.

Although the present invention and its advantages have been described indetail, it should be understood that various changes, substitutions andalternations can be made herein without departing from the spirit andscope of the invention as defined by the appended claims.

1. A method of estimating electrical parameters of an earth formationusing a measuring device disposed to be deployed in a borehole, themeasuring device disposed to collect formation data, the measuringdevice having a pre-established model associated therewith, thepre-established model governing the processing of the formation data todetermine electrical parameters regarding the earth formation, themethod comprising: (a) providing a simplified model associated with themeasuring device, the simplified model making at least one simplifyingassumption about the pre-established model; (b) receiving a set ofcollected formation data regarding the earth formation, the set ofcollected formation data collected by the measuring device; (c)normalizing the set of collected formation data to derive a renormalizeddataset, wherein the renormalized dataset is substantially consistentwith each of the simplifying assumptions in (a); (d) applying therenormalized dataset to the simplified model to estimate at least oneelectrical parameter of the earth formation; and (e) using the estimateof the at least one electrical parameter of the earth formation in amanner selected from the group consisting of detecting a hydrocarbonwithin the earth formation, guiding a drill bit within a productive zoneof the earth formation, estimating poor pressure of the earth formation,evaluating geological features of the earth formation, and detectingvertical fractures within the earth formation.
 2. The method of claim 1,wherein at least one of the electrical parameters estimated in (d) isselected from the group consisting of conductivity and dielectricconstant.
 3. The method of claim 1, in which the measurement devicecomprises a mandrel, and wherein at least one of the simplifyingassumptions in (a) includes a simplified assumption with respect to theeffect of the mandrel on the pre-established model.
 4. The method ofclaim 1, in which the measurement device comprises a transmitter antennaof finite size, and wherein at least one of the simplifying assumptionsin (a) includes an assumption regarding the size of an antenna in thesimplified model.
 5. The method of claim 4, wherein the assumptionregarding the size of an antenna in the simplified model is that saidantenna is infinitesimal.
 6. The method of claim 1, wherein themeasuring device collects formation data via analysis of an electricsignal transmitted by the measuring device.
 7. The method of claim 6,wherein the set of collected formation data includes data representingat least one characteristic of the electric signal selected from thegroup consisting of attenuation and phase shift.
 8. The method of claim1, wherein the set of collected formation data includes data that may berepresented as a complex number.
 9. The method of claim 1, wherein therenormalized dataset includes data that represent point-dipole values.10. The method of claim 1, wherein (c) includes transforming the set ofcollected formation data into the renormalized dataset according to atleast one member of the group consisting of: (1) at least onepredetermined equation; (2) a predefined multi-dimensional look-uptable; and (3) a predefined one-dimensional look-up table.
 11. Themethod of claim 1, wherein (c) includes transforming the set ofcollected formation data into the renormalized dataset according toweighted sums, the weighted sums comprising: $\begin{matrix}{g_{a}^{\prime} = {\sum\limits_{i = 1}^{N}{c_{ia}{f_{ia}\left( g_{a} \right)}}}} \\{g_{p}^{\prime} = {\sum\limits_{i = 1}^{N}{c_{ip}{f_{ip}\left( g_{p} \right)}}}}\end{matrix}$ where g_(a) and g_(p) represent collected formation data,g′_(a) and g′_(p) represent corresponding members of the renormalizeddataset, i=1 to N terms in the weighted sums, ƒ_(ia) and ƒ_(ip) arefunctions representing preselected transformations between mandrel andpoint-dipole data over prescribed trajectories in the g_(a), g_(p)plane, and c_(ia) and c_(ip) are coefficients chosen to produce valuesfor g′_(a) and g′_(p) that approximate point-dipole values over ananticipated range of g_(a) and g_(p).
 12. The method of claim 10,wherein the one-dimensional look-up table comprises: a plurality of setsof table data, each set of table data matched to a corresponding one ofa plurality of hypothetical earth formations, each set of table datafurther including a predicted measurement and a correspondingrenormalized measurement that are each indicative of an electricalparameter of the corresponding hypothetical earth formation, whereintransformation from each predicted measurement to its correspondingrenormalized measurement substantially reflects application of one ofthe simplifying assumptions to the predicted measurement.
 13. The methodof claim 12, wherein (c) in claim 1 further comprises deriving membersof the renormalized dataset by interpolating members of the set ofcollected formation data between sets of table data in theone-dimensional look-up table.
 14. The method of claim 12, wherein theone-dimensional look up table was previously generated by deriving eachof the plurality of sets of table data for a single dielectric constantvalue and a series of corresponding resistivity values.
 15. The methodof claim 12, wherein the one-dimensional look up table was previouslygenerated by deriving each of the plurality of sets of table data for asingle resistivity value and a series of corresponding dielectricconstant values.
 16. The method of claim 12, wherein each of apredetermined subset of the plurality of hypothetical earth formationsassumes a selected dielectric constant value and one of a plurality ofdifferent corresponding conductivity values.
 17. The method of claim 12,wherein the plurality of hypothetical earth formations includes avacuum.
 18. The method of claim 1, wherein the simplified model includesa calibration factor to adjust for anomalies in the measurement device.19. The method of claim 1, wherein the simplified model includes acomponent for adjusting for effects on the formation data caused by theborehole.
 20. A processor-readable medium on which processor-executablelogic may be stored, the processor-executable logic disposed to generateinstructions to a computer processor, the instructions disposed to causethe computer processor to follow a method of estimating electricalparameters of an earth formation using a measuring device disposed to bedeployed in a borehole, the measuring device disposed to collectformation data, the measuring device having a pre-established modelassociated therewith, the pre-established model governing the processingof the formation data to determine electrical parameters regarding theearth formation, method comprising: (a) providing a simplified modelassociated with the measuring device, the simplified model making atleast one simplifying assumption about the pre-established model; (b)receiving a set of collected formation data regarding the earthformation, the set of collected formation data collected by themeasuring device; (c) normalizing the set of collected formation data toderive a renormalized dataset, wherein the renormalized dataset issubstantially consistent with each of the simplifying assumptions in(a); (d) applying the renormalized dataset to the simplified model toestimate at least one electrical parameter of the earth and formation;and (e) using the estimate of the at least one electrical parameter ofthe earth formation in a manner selected from the group consisting ofdetecting a hydrocarbon within the earth formation, guiding a drill bitwithin a productive zone of the earth formation, estimating poorpressure of the earth formation, evaluating geological features of theearth formation, and detecting vertical fractures within the earthformation.
 21. The processor-readable medium of claim 20, in which themeasurement device comprises a mandrel, and wherein at least one of thesimplifying assumptions in (a) includes a simplified assumption withrespect to the effect of the mandrel on the pre-established model. 22.The processor-readable medium of claim 20, in which the measurementdevice comprises a transmitter antenna of finite size, and wherein atleast one of the simplifying assumptions in (a) includes an assumptionregarding the size of an antenna in the simplified model.
 23. Theprocessor-readable medium of claim 22, wherein the assumption regardingthe size of an antenna in the simplified model is that said antenna isinfinitesimal.
 24. The processor-readable medium of claim 20, whereinthe measuring device collects formation data via analysis of an electricsignal transmitted by the measuring device.
 25. The processor-readablemedium of claim 20, wherein the renormalized dataset includes data thatrepresent point-dipole values.
 26. The method of claim 20, wherein (c)includes transforming the set of collected formation data into therenormalized dataset according to at least one member of the groupconsisting of: (1) at least one predetermined equation; (2) a predefinedmulti-dimensional look-up table; and (3) a predefined one-dimensionallook-up table.
 27. The method of claim 20, wherein (c) includestransforming the set of collected formation data into the renormalizeddataset according to weighted sums, the weighted sums comprising:$\begin{matrix}{g_{a}^{\prime} = {\sum\limits_{i = 1}^{N}{c_{ia}{f_{ia}\left( g_{a} \right)}}}} \\{g_{p}^{\prime} = {\sum\limits_{i = 1}^{N}{c_{ip}{f_{ip}\left( g_{p} \right)}}}}\end{matrix}$ where g_(a) and g_(p) represent collected formation data,g′_(a) and g′_(p) represent corresponding members of the renormalizeddataset, i=1 to N terms in the weighted sums, ƒ_(ia) and ƒ_(ip) arefunctions representing preselected transformations between mandrel andpoint-dipole data over prescribed trajectories in the g_(a), g_(p)plane, and c_(ia) and c_(ip) are coefficients chosen to produce valuesfor g′_(a) and g′_(p) that approximate point-dipole values over ananticipated range of g_(a) and g_(p).
 28. The processor-readable mediumof claim 26, wherein the one-dimensional look-up table comprises: aplurality of sets of table data, each set of table data matched to acorresponding one of a plurality of hypothetical earth formations, eachset of table data further including a predicted measurement and acorresponding renormalized measurement that are each indicative of anelectrical parameter of the corresponding hypothetical earth formation,wherein transformation from each predicted measurement to itscorresponding renormalized measurement substantially reflectsapplication of one of the simplifying assumptions to the predictedmeasurement.
 29. The processor-readable medium of claim 26, wherein (c)in claim 20 further comprises deriving members of the renormalizeddataset by interpolating members of the set of collected formation databetween sets of table data in the one-dimensional look-up table.
 30. Theprocessor-readable medium of claim 26 wherein the one-dimensional lookup table was previously generated by deriving each of the plurality ofsets of table data for a single dielectric constant value and a seriesof corresponding resistivity values.
 31. The processor-readable mediumof claim 26, wherein the one-dimensional look up table was previouslygenerated by deriving each of the plurality of sets of table data for asingle resistivity value and a series of corresponding dielectricconstant values.
 32. The processor-readable medium of claim 26, whereineach of a predetermined subset of the plurality of hypothetical earthformations assumes a selected dielectric constant value and one of aplurality of different corresponding conductivity values.
 33. A methodof estimating electrical parameters of an earth formation using ameasuring device disposed to be deployed in a borehole, the measuringdevice disposed to collect formation data via analysis of an electricalsignal transmitted by the measuring device, the measuring device havinga pre-established model associated therewith, the pre-established modelgoverning the processing of the formation data to determine electricalparameters regarding the earth formation, the method comprising: (a)providing a simplified model associated with the measuring device, thesimplified model making at least one simplifying assumption about thepre-established model; (b) receiving a set of collected formation dataregarding the earth formation, the set of collected formation datacollected by the measuring device; (c) normalizing the set of collectedformation data to derive a renormalized dataset, wherein therenormalized dataset is substantially consistent with each of thesimplifying assumptions in (a), said normalizing including transformingthe set of collected formation data into the renormalized datasetaccording to a predefined one-dimensional look-up table comprising aplurality of sets of table data, each set of table data matched to acorresponding one of a plurality of hypothetical earth formations, eachset of table data further including a predicted measurement and acorresponding renormalized measurement that are each indicative of anelectrical parameter of the corresponding hypothetical earth formation,wherein transformation from each predicted measurement to itscorresponding renormalized measurement substantially reflectsapplication of one of the simplifying assumptions to the predictedmeasurement; (d) applying the renormalized dataset to the simplifiedmodel to estimate at least one electrical parameter of the earthformation; and (e) using the estimate of the at least one electricalparameter of the earth formation in a manner selected from the groupconsisting of detecting a hydrocarbon within the earth formation,guiding a drill bit within a productive zone of the earth formation,estimating poor pressure of the earth formation, evaluating geologicalfeatures of the earth formation, and detecting vertical fractures withinthe earth formation.
 34. The method of claim 33, in which themeasurement device comprises a mandrel, and wherein at least one of thesimplifying assumptions in (a) includes a simplified assumption withrespect to the effect of the mandrel on the pre-established model. 35.The method of claim 33, in which the measurement device comprises atransmitter antenna of finite size, and wherein at least one of thesimplifying assumptions in (a) includes an assumption regarding the sizeof an antenna in the simplified model.
 36. The method of claim 35,wherein the assumption regarding the size of an antenna in thesimplified model is that said antenna is infinitesimal.
 37. The methodof claim 33, wherein the renormalized dataset includes data thatrepresent point-dipole values.
 38. The method of claim 33, wherein (c)further comprises deriving members of the renormalized dataset byinterpolating members of the set of collected formation data betweensets of table data in the one-dimensional look-up table.
 39. The methodof claim 33 wherein the one-dimensional look up table was previouslygenerated by deriving each of the plurality of sets of table data for asingle dielectric constant value and a series of correspondingresistivity values.
 40. The method of claim 33, wherein theone-dimensional look up table was previously generated by deriving eachof the plurality of sets of table data for a single resistivity valueand a series of corresponding dielectric constant values.
 41. The methodof claim 33, wherein each of a predetermined subset of the plurality ofhypothetical earth formations assumes a selected dielectric constantvalue and one of a plurality of different corresponding conductivityvalues.
 42. A method of estimating electrical parameters of an earthformation using a measuring device disposed to be deployed in aborehole, the measuring device disposed to collect formation data, themeasuring device having a pre-established model associated therewith,the pre-established model governing the processing of the formation datato determine electrical parameters regarding the earth formation, themethod comprising: (a) providing a simplified model associated with themeasuring device, the simplified model making at least one simplifyingassumption about the pre-established model; (b) receiving a set ofcollected formation data regarding the earth formation, the set ofcollected formation data collected by the measuring device; (c)normalizing the set of collected formation data to derive a renormalizeddataset, wherein the renormalized dataset is substantially consistentwith each of the simplifying assumptions in (a), said normalizingincluding transforming the set of collected formation data into therenormalized dataset according to weighted sums, the weighted sumscomprising: $\begin{matrix}{g_{a}^{\prime} = {\sum\limits_{i = 1}^{N}{c_{ia}{f_{ia}\left( g_{a} \right)}}}} \\{g_{p}^{\prime} = {\sum\limits_{i = 1}^{N}{c_{ip}{f_{ip}\left( g_{p} \right)}}}}\end{matrix}$  where g_(a) and g_(p) represent collected formation data,g′_(a) and g′_(p) represent corresponding members of the renormalizeddataset, i=1 to N terms in the weighted sums, ƒ_(ia) and ƒ_(ip) arefunctions representing preselected transformations between mandrel andpoint-dipole data over prescribed trajectories in the g_(a), g_(p)plane, and c_(ia) and c_(ip) are coefficients chosen to produce valuesfor g′_(a) and g′_(p) that approximate point-dipole values over ananticipated range of g_(a) and g_(p); (d) applying the renormalizeddataset to the simplified model to estimate at least one electricalparameter of the earth formation; and (e) using the estimate of the atleast one electrical parameter of the earth formation in a mannerselected from the group consisting of detecting a hydrocarbon within theearth formation, guiding a drill bit within a productive zone of theearth formation, estimating poor pressure of the earth formation,evaluating geological features of the earth formation, and detectingvertical fractures within the earth formation.
 43. The method of claim42, in which the measurement device comprises a mandrel, and wherein atleast one of the simplifying assumptions in (a) includes a simplifiedassumption with respect to the effect of the mandrel on thepre-established model.
 44. The method of claim 42, in which themeasurement device comprises a transmitter antenna of finite size, andwherein at least one of the simplifying assumptions in (a) includes anassumption regarding the size of an antenna in the simplified model. 45.The method of claim 44, wherein the assumption regarding the size of anantenna in the simplified model is that said antenna is infinitesimal.